Dynamic programming

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https://en.wikipedia.org/wiki/Dynamic_programming 

 

Figure 1. Finding the shortest path in a graph using optimal substructure; a straight line indicates a single edge; a wavy line indicates a shortest path between the two vertices it connects (among other paths, not shown, sharing the same two vertices); the bold line is the overall shortest path from start to goal.

Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics.

In both contexts it refers to simplifying a complicated problem by breaking it down into simpler sub-problems in a recursive manner. While some decision problems cannot be taken apart this way, decisions that span several points in time do often break apart recursively. Likewise, in computer science, if a problem can be solved optimally by breaking it into sub-problems and then recursively finding the optimal solutions to the sub-problems, then it is said to have optimal substructure.

If sub-problems can be nested recursively inside larger problems, so that dynamic programming methods are applicable, then there is a relation between the value of the larger problem and the values of the sub-problems. In the optimization literature this relationship is called the Bellman equation.

Overview

Mathematical optimization

In terms of mathematical optimization, dynamic programming usually refers to simplifying a decision by breaking it down into a sequence of decision steps over time. This is done by defining a sequence of value functions V₁, V₂, ..., V_n taking y as an argument representing the state of the system at times i from 1 to n. The definition of V_n(y) is the value obtained in state y at the last time n. The values V_i at earlier times i = n −1, n − 2, ..., 2, 1 can be found by working backwards, using a recursive relationship called the Bellman equation. For i = 2, ..., n, V_i−1 at any state y is calculated from V_i by maximizing a simple function (usually the sum) of the gain from a decision at time i − 1 and the function V_i at the new state of the system if this decision is made. Since V_i has already been calculated for the needed states, the above operation yields V_i−1 for those states. Finally, V₁ at the initial state of the system is the value of the optimal solution. The optimal values of the decision variables can be recovered, one by one, by tracking back the calculations already performed.

Control theory

In control theory, a typical problem is to find an admissible control ${\displaystyle {\mathbf{u}}^{\ast }}$ which causes the system ${\displaystyle \dot{\mathbf{x}}(t)=\mathbf{g}\left(\mathbf{x}(t),\mathbf{u}(t),t\right)}$ to follow an admissible trajectory ${\displaystyle {\mathbf{x}}^{\ast }}$ on a continuous time interval ${\displaystyle t_0\le t\le t_1}$ that minimizes a cost function

${\displaystyle J=b\left(\mathbf{x}(t_1),t_1\right)+{\int }_{t_0}^{t_1}f\left(\mathbf{x}(t),\mathbf{u}(t),t\right)\mathrm{d}t}$

The solution to this problem is an optimal control law or policy ${\displaystyle {\mathbf{u}}^{\ast }=h(\mathbf{x}(t),t)}$, which produces an optimal trajectory ${\displaystyle {\mathbf{x}}^{\ast }}$ and a cost-to-go function ${\displaystyle J^{\ast }}$. The latter obeys the fundamental equation of dynamic programming:

${\displaystyle -J_t^{\ast }=\underset{\mathbf{u}}{min}\left\{f\left(\mathbf{x}(t),\mathbf{u}(t),t\right)+J_x^{\ast \mathsf{T}}\mathbf{g}\left(\mathbf{x}(t),\mathbf{u}(t),t\right)\right\}}$

a partial differential equation known as the Hamilton–Jacobi–Bellman equation, in which ${\displaystyle J_x^{\ast }=\frac{\mathrm{\partial }{J}^{\ast }}{\mathrm{\partial }\mathbf{x}}={\left[\frac{\mathrm{\partial }{J}^{\ast }}{\mathrm{\partial }{x}_1}\frac{\mathrm{\partial }{J}^{\ast }}{\mathrm{\partial }{x}_2}\dots \frac{\mathrm{\partial }{J}^{\ast }}{\mathrm{\partial }{x}_n}\right]}^{\mathsf{T}}}$ and ${\displaystyle J_t^{\ast }=\frac{\mathrm{\partial }{J}^{\ast }}{\mathrm{\partial }t}}$. One finds that minimizing ${\displaystyle \mathbf{u}}$ in terms of ${\displaystyle t}$, ${\displaystyle \mathbf{x}}$, and the unknown function ${\displaystyle J_x^{\ast }}$ and then substitutes the result into the Hamilton–Jacobi–Bellman equation to get the partial differential equation to be solved with boundary condition ${\displaystyle J\left(t_1\right)=b\left(\mathbf{x}(t_1),t_1\right)}$. In practice, this generally requires numerical techniques for some discrete approximation to the exact optimization relationship.

Alternatively, the continuous process can be approximated by a discrete system, which leads to a following recurrence relation analog to the Hamilton–Jacobi–Bellman equation:

${\displaystyle J_k^{\ast }\left({\mathbf{x}}_{n-k}\right)=\underset{{\mathbf{u}}_{n-k}}{min}\left\{\hat{f}\left({\mathbf{x}}_{n-k},{\mathbf{u}}_{n-k}\right)+J_{k-1}^{\ast }\left(\hat{\mathbf{g}}\left({\mathbf{x}}_{n-k},{\mathbf{u}}_{n-k}\right)\right)\right\}}$

at the ${\displaystyle k}$-th stage of ${\displaystyle n}$ equally spaced discrete time intervals, and where ${\displaystyle \hat{f}}$ and ${\displaystyle \hat{\mathbf{g}}}$ denote discrete approximations to ${\displaystyle f}$ and ${\displaystyle \mathbf{g}}$. This functional equation is known as the Bellman equation, which can be solved for an exact solution of the discrete approximation of the optimization equation.

Example from economics: Ramsey's problem of optimal saving

See also: Ramsey–Cass–Koopmans model

In economics, the objective is generally to maximize (rather than minimize) some dynamic social welfare function. In Ramsey's problem, this function relates amounts of consumption to levels of utility. Loosely speaking, the planner faces the trade-off between contemporaneous consumption and future consumption (via investment in capital stock that is used in production), known as intertemporal choice. Future consumption is discounted at a constant rate ${\displaystyle \beta \in (0,1)}$. A discrete approximation to the transition equation of capital is given by

${\displaystyle k_{t+1}=\hat{g}\left(k_t,c_t\right)=f(k_t)-c_t}$

where ${\displaystyle c}$ is consumption, ${\displaystyle k}$ is capital, and ${\displaystyle f}$ is a production function satisfying the Inada conditions. An initial capital stock ${\displaystyle k_0>0}$ is assumed.

Let ${\displaystyle c_t}$ be consumption in period t, and assume consumption yields utility ${\displaystyle u(c_t)=\ln (c_t)}$ as long as the consumer lives. Assume the consumer is impatient, so that he discounts future utility by a factor b each period, where ${\displaystyle 0<b<1}$. Let ${\displaystyle k_t}$ be capital in period t. Assume initial capital is a given amount ${\displaystyle k_0>0}$, and suppose that this period's capital and consumption determine next period's capital as ${\displaystyle k_{t+1}=A{k}_t^a-c_t}$, where A is a positive constant and ${\displaystyle 0<a<1}$. Assume capital cannot be negative. Then the consumer's decision problem can be written as follows:

${\displaystyle max\sum _{t=0}^T{b}^t\ln (c_t)}$ subject to ${\displaystyle k_{t+1}=A{k}_t^a-c_t\ge 0}$ for all ${\displaystyle t=0,1,2,\dots ,T}$

Written this way, the problem looks complicated, because it involves solving for all the choice variables ${\displaystyle c_0,c_1,c_2,\dots ,c_T}$. (The capital ${\displaystyle k_0}$ is not a choice variable—the consumer's initial capital is taken as given.)

The dynamic programming approach to solve this problem involves breaking it apart into a sequence of smaller decisions. To do so, we define a sequence of value functions ${\displaystyle V_t(k)}$, for ${\displaystyle t=0,1,2,\dots ,T,T+1}$ which represent the value of having any amount of capital k at each time t. There is (by assumption) no utility from having capital after death, ${\displaystyle V_{T+1}(k)=0}$.

The value of any quantity of capital at any previous time can be calculated by backward induction using the Bellman equation. In this problem, for each ${\displaystyle t=0,1,2,\dots ,T}$, the Bellman equation is

${\displaystyle V_t(k_t)=max\left(\ln (c_t)+b{V}_{t+1}(k_{t+1})\right)}$ subject to ${\displaystyle k_{t+1}=A{k}_t^a-c_t\ge 0}$

This problem is much simpler than the one we wrote down before, because it involves only two decision variables, ${\displaystyle c_t}$ and ${\displaystyle k_{t+1}}$. Intuitively, instead of choosing his whole lifetime plan at birth, the consumer can take things one step at a time. At time t, his current capital ${\displaystyle k_t}$ is given, and he only needs to choose current consumption ${\displaystyle c_t}$ and saving ${\displaystyle k_{t+1}}$.

To actually solve this problem, we work backwards. For simplicity, the current level of capital is denoted as k. ${\displaystyle V_{T+1}(k)}$ is already known, so using the Bellman equation once we can calculate ${\displaystyle V_T(k)}$, and so on until we get to ${\displaystyle V_0(k)}$, which is the value of the initial decision problem for the whole lifetime. In other words, once we know ${\displaystyle V_{T-j+1}(k)}$, we can calculate ${\displaystyle V_{T-j}(k)}$, which is the maximum of ${\displaystyle \ln (c_{T-j})+b{V}_{T-j+1}(A{k}^a-c_{T-j})}$, where ${\displaystyle c_{T-j}}$ is the choice variable and ${\displaystyle A{k}^a-c_{T-j}\ge 0}$.

Working backwards, it can be shown that the value function at time ${\displaystyle t=T-j}$ is

${\displaystyle V_{T-j}(k)=a\sum _{i=0}^j{a}^i{b}^i\ln k+v_{T-j}}$

where each ${\displaystyle v_{T-j}}$ is a constant, and the optimal amount to consume at time ${\displaystyle t=T-j}$ is

${\displaystyle c_{T-j}(k)=\frac{1}{\sum _{i=0}^j{a}^i{b}^i}A{k}^a}$

which can be simplified to

${\displaystyle \begin{array}{rl}{c}_T(k)& =A{k}^a\\ c_{T-1}(k)& =\frac{A{k}^a}{1+ab}\\ c_{T-2}(k)& =\frac{A{k}^a}{1+ab+a^2{b}^2}\\ & \dots \\ c_2(k)& =\frac{A{k}^a}{1+ab+a^2{b}^2+\dots +a^{T-2}{b}^{T-2}}\\ c_1(k)& =\frac{A{k}^a}{1+ab+a^2{b}^2+\dots +a^{T-2}{b}^{T-2}+a^{T-1}{b}^{T-1}}\\ c_0(k)& =\frac{A{k}^a}{1+ab+a^2{b}^2+\dots +a^{T-2}{b}^{T-2}+a^{T-1}{b}^{T-1}+a^T{b}^T}\end{array}}$

We see that it is optimal to consume a larger fraction of current wealth as one gets older, finally consuming all remaining wealth in period T, the last period of life.

Computer programming

There are two key attributes that a problem must have in order for dynamic programming to be applicable: optimal substructure and overlapping sub-problems. If a problem can be solved by combining optimal solutions to non-overlapping sub-problems, the strategy is called "divide and conquer" instead. This is why merge sort and quick sort are not classified as dynamic programming problems.

Optimal substructure means that the solution to a given optimization problem can be obtained by the combination of optimal solutions to its sub-problems. Such optimal substructures are usually described by means of recursion. For example, given a graph G=(V,E), the shortest path p from a vertex u to a vertex v exhibits optimal substructure: take any intermediate vertex w on this shortest path p. If p is truly the shortest path, then it can be split into sub-paths p₁ from u to w and p₂ from w to v such that these, in turn, are indeed the shortest paths between the corresponding vertices (by the simple cut-and-paste argument described in Introduction to Algorithms). Hence, one can easily formulate the solution for finding shortest paths in a recursive manner, which is what the Bellman–Ford algorithm or the Floyd–Warshall algorithm does.

Overlapping sub-problems means that the space of sub-problems must be small, that is, any recursive algorithm solving the problem should solve the same sub-problems over and over, rather than generating new sub-problems. For example, consider the recursive formulation for generating the Fibonacci series: F_i = F_i−1 + F_i−2, with base case F₁ = F₂ = 1. Then F₄₃ = F₄₂ + F₄₁, and F₄₂ = F₄₁ + F₄₀. Now F₄₁ is being solved in the recursive sub-trees of both F₄₃ as well as F₄₂. Even though the total number of sub-problems is actually small (only 43 of them), we end up solving the same problems over and over if we adopt a naive recursive solution such as this. Dynamic programming takes account of this fact and solves each sub-problem only once.

Figure 2. The subproblem graph for the Fibonacci sequence. The fact that it is not a tree indicates overlapping subproblems.

This can be achieved in either of two ways:

• Top-down approach: This is the direct fall-out of the recursive formulation of any problem. If the solution to any problem can be formulated recursively using the solution to its sub-problems, and if its sub-problems are overlapping, then one can easily memoize or store the solutions to the sub-problems in a table. Whenever we attempt to solve a new sub-problem, we first check the table to see if it is already solved. If a solution has been recorded, we can use it directly, otherwise we solve the sub-problem and add its solution to the table.
• Bottom-up approach: Once we formulate the solution to a problem recursively as in terms of its sub-problems, we can try reformulating the problem in a bottom-up fashion: try solving the sub-problems first and use their solutions to build-on and arrive at solutions to bigger sub-problems. This is also usually done in a tabular form by iteratively generating solutions to bigger and bigger sub-problems by using the solutions to small sub-problems. For example, if we already know the values of F₄₁ and F₄₀, we can directly calculate the value of F₄₂.

Some programming languages can automatically memoize the result of a function call with a particular set of arguments, in order to speed up call-by-name evaluation (this mechanism is referred to as call-by-need). Some languages make it possible portably (e.g. Scheme, Common Lisp, Perl or D). Some languages have automatic memoization built in, such as tabled Prolog and J, which supports memoization with the M. adverb. In any case, this is only possible for a referentially transparent function. Memoization is also encountered as an easily accessible design pattern within term-rewrite based languages such as Wolfram Language.

Bioinformatics

Dynamic programming is widely used in bioinformatics for tasks such as sequence alignment, protein folding, RNA structure prediction and protein-DNA binding. The first dynamic programming algorithms for protein-DNA binding were developed in the 1970s independently by Charles DeLisi in USA and Georgii Gurskii and Alexander Zasedatelev in USSR. Recently these algorithms have become very popular in bioinformatics and computational biology, particularly in the studies of nucleosome positioning and transcription factor binding.

Examples: computer algorithms

Dijkstra's algorithm for the shortest path problem

From a dynamic programming point of view, Dijkstra's algorithm for the shortest path problem is a successive approximation scheme that solves the dynamic programming functional equation for the shortest path problem by the Reaching method.

In fact, Dijkstra's explanation of the logic behind the algorithm, namely

Problem 2. Find the path of minimum total length between two given nodes ${\displaystyle P}$ and ${\displaystyle Q}$.

We use the fact that, if ${\displaystyle R}$ is a node on the minimal path from ${\displaystyle P}$ to ${\displaystyle Q}$, knowledge of the latter implies the knowledge of the minimal path from ${\displaystyle P}$ to ${\displaystyle R}$.

is a paraphrasing of Bellman's famous Principle of Optimality in the context of the shortest path problem.

Fibonacci sequence

Using dynamic programming in the calculation of the nth member of the Fibonacci sequence improves its performance greatly. Here is a naïve implementation, based directly on the mathematical definition:

function fib(n)
    if n <= 1 return n
    return fib(n − 1) + fib(n − 2)

Notice that if we call, say, fib(5), we produce a call tree that calls the function on the same value many different times:

1. fib(5)
2. fib(4) + fib(3)
3. (fib(3) + fib(2)) + (fib(2) + fib(1))
4. ((fib(2) + fib(1)) + (fib(1) + fib(0))) + ((fib(1) + fib(0)) + fib(1))
5. (((fib(1) + fib(0)) + fib(1)) + (fib(1) + fib(0))) + ((fib(1) + fib(0)) + fib(1))

In particular, fib(2) was calculated three times from scratch. In larger examples, many more values of fib, or subproblems, are recalculated, leading to an exponential time algorithm.

Now, suppose we have a simple map object, m, which maps each value of fib that has already been calculated to its result, and we modify our function to use it and update it. The resulting function requires only O(n) time instead of exponential time (but requires O(n) space):

var m := map(0 → 0, 1 → 1)
function fib(n)
    if key n is not in map m 
        m[n] := fib(n − 1) + fib(n − 2)
    return m[n]

This technique of saving values that have already been calculated is called memoization; this is the top-down approach, since we first break the problem into subproblems and then calculate and store values.

In the bottom-up approach, we calculate the smaller values of fib first, then build larger values from them. This method also uses O(n) time since it contains a loop that repeats n − 1 times, but it only takes constant (O(1)) space, in contrast to the top-down approach which requires O(n) space to store the map.

function fib(n)
    if n = 0
        return 0
    else
        var previousFib := 0, currentFib := 1
        repeat n − 1 times // loop is skipped if n = 1
            var newFib := previousFib + currentFib
            previousFib := currentFib
            currentFib  := newFib
        return currentFib

In both examples, we only calculate fib(2) one time, and then use it to calculate both fib(4) and fib(3), instead of computing it every time either of them is evaluated.

The above method actually takes ${\displaystyle \mathrm{\Omega }(n^2)}$ time for large n because addition of two integers with ${\displaystyle \mathrm{\Omega }(n)}$ bits each takes ${\displaystyle \mathrm{\Omega }(n)}$ time. (The n^th fibonacci number has ${\displaystyle \mathrm{\Omega }(n)}$ bits.) Also, there is a closed form for the Fibonacci sequence, known as Binet's formula, from which the ${\displaystyle n}$-th term can be computed in approximately ${\displaystyle O(n(\log n{)}^2)}$ time, which is more efficient than the above dynamic programming technique. However, the simple recurrence directly gives the matrix form that leads to an approximately ${\displaystyle O(n\log n)}$ algorithm by fast matrix exponentiation.

A type of balanced 0–1 matrix

Consider the problem of assigning values, either zero or one, to the positions of an n × n matrix, with n even, so that each row and each column contains exactly n / 2 zeros and n / 2 ones. We ask how many different assignments there are for a given ${\displaystyle n}$. For example, when n = 4, five possible solutions are

${\displaystyle \left[\begin{array}{cccc}0& 1& 0& 1\\ 1& 0& 1& 0\\ 0& 1& 0& 1\\ 1& 0& 1& 0\end{array}\right]\text{ and }\left[\begin{array}{cccc}0& 0& 1& 1\\ 0& 0& 1& 1\\ 1& 1& 0& 0\\ 1& 1& 0& 0\end{array}\right]\text{ and }\left[\begin{array}{cccc}1& 1& 0& 0\\ 0& 0& 1& 1\\ 1& 1& 0& 0\\ 0& 0& 1& 1\end{array}\right]\text{ and }\left[\begin{array}{cccc}1& 0& 0& 1\\ 0& 1& 1& 0\\ 0& 1& 1& 0\\ 1& 0& 0& 1\end{array}\right]\text{ and }\left[\begin{array}{cccc}1& 1& 0& 0\\ 1& 1& 0& 0\\ 0& 0& 1& 1\\ 0& 0& 1& 1\end{array}\right].}$

There are at least three possible approaches: brute force, backtracking, and dynamic programming.

Brute force consists of checking all assignments of zeros and ones and counting those that have balanced rows and columns (n / 2 zeros and n / 2 ones). As there are ${\displaystyle 2^{n^2}}$ possible assignments and ${\displaystyle {(\genfrac{}{}{0ex}{}{n}{n/2})}^n}$ sensible assignments, this strategy is not practical except maybe up to ${\displaystyle n=6}$.

Backtracking for this problem consists of choosing some order of the matrix elements and recursively placing ones or zeros, while checking that in every row and column the number of elements that have not been assigned plus the number of ones or zeros are both at least n / 2. While more sophisticated than brute force, this approach will visit every solution once, making it impractical for n larger than six, since the number of solutions is already 116,963,796,250 for n = 8, as we shall see.

Dynamic programming makes it possible to count the number of solutions without visiting them all. Imagine backtracking values for the first row – what information would we require about the remaining rows, in order to be able to accurately count the solutions obtained for each first row value? We consider k × n boards, where 1 ≤ k ≤ n, whose ${\displaystyle k}$ rows contain ${\displaystyle n/2}$ zeros and ${\displaystyle n/2}$ ones. The function f to which memoization is applied maps vectors of n pairs of integers to the number of admissible boards (solutions). There is one pair for each column, and its two components indicate respectively the number of zeros and ones that have yet to be placed in that column. We seek the value of ${\displaystyle f((n/2,n/2),(n/2,n/2),\dots (n/2,n/2))}$ (${\displaystyle n}$ arguments or one vector of ${\displaystyle n}$ elements). The process of subproblem creation involves iterating over every one of ${\displaystyle (\genfrac{}{}{0ex}{}{n}{n/2})}$ possible assignments for the top row of the board, and going through every column, subtracting one from the appropriate element of the pair for that column, depending on whether the assignment for the top row contained a zero or a one at that position. If any one of the results is negative, then the assignment is invalid and does not contribute to the set of solutions (recursion stops). Otherwise, we have an assignment for the top row of the k × n board and recursively compute the number of solutions to the remaining (k − 1) × n board, adding the numbers of solutions for every admissible assignment of the top row and returning the sum, which is being memoized. The base case is the trivial subproblem, which occurs for a 1 × n board. The number of solutions for this board is either zero or one, depending on whether the vector is a permutation of n / 2 ${\displaystyle (0,1)}$ and n / 2 ${\displaystyle (1,0)}$ pairs or not.

For example, in the first two boards shown above the sequences of vectors would be

((2, 2) (2, 2) (2, 2) (2, 2))       ((2, 2) (2, 2) (2, 2) (2, 2))     k = 4
  0      1      0      1              0      0      1      1

((1, 2) (2, 1) (1, 2) (2, 1))       ((1, 2) (1, 2) (2, 1) (2, 1))     k = 3
  1      0      1      0              0      0      1      1

((1, 1) (1, 1) (1, 1) (1, 1))       ((0, 2) (0, 2) (2, 0) (2, 0))     k = 2
  0      1      0      1              1      1      0      0

((0, 1) (1, 0) (0, 1) (1, 0))       ((0, 1) (0, 1) (1, 0) (1, 0))     k = 1
  1      0      1      0              1      1      0      0

((0, 0) (0, 0) (0, 0) (0, 0))       ((0, 0) (0, 0), (0, 0) (0, 0))

The number of solutions (sequence A058527 in the OEIS) is

${\displaystyle 1,2,90,297200,116963796250,6736218287430460752,\dots }$

Links to the MAPLE implementation of the dynamic programming approach may be found among the external links.

Checkerboard

Consider a checkerboard with n × n squares and a cost function c(i, j) which returns a cost associated with square (i,j) (i being the row, j being the column). For instance (on a 5 × 5 checkerboard),

5	6	7	4	7	8
4	7	6	1	1	4
3	3	5	7	8	2
2	–	6	7	0	–
1	–	–	*5*	–	–
	1	2	3	4	5

Thus c(1, 3) = 5

Let us say there was a checker that could start at any square on the first rank (i.e., row) and you wanted to know the shortest path (the sum of the minimum costs at each visited rank) to get to the last rank; assuming the checker could move only diagonally left forward, diagonally right forward, or straight forward. That is, a checker on (1,3) can move to (2,2), (2,3) or (2,4).

5
4
3
2		x	x	x
1			o
	1	2	3	4	5

This problem exhibits optimal substructure. That is, the solution to the entire problem relies on solutions to subproblems. Let us define a function q(i, j) as

q(i, j) = the minimum cost to reach square (i, j).

Starting at rank n and descending to rank 1, we compute the value of this function for all the squares at each successive rank. Picking the square that holds the minimum value at each rank gives us the shortest path between rank n and rank 1.

The function q(i, j) is equal to the minimum cost to get to any of the three squares below it (since those are the only squares that can reach it) plus c(i, j). For instance:

5
4			A
3		B	C	D
2
1
	1	2	3	4	5

${\displaystyle q(A)=min(q(B),q(C),q(D))+c(A)}$

Now, let us define q(i, j) in somewhat more general terms:

${\displaystyle q(i,j)=\{\begin{array}{ll}\mathrm{\infty }& j<1\text{ or }j>n\\ c(i,j)& i=1\\ min(q(i-1,j-1),q(i-1,j),q(i-1,j+1))+c(i,j)& \text{otherwise.}\end{array}}$

The first line of this equation deals with a board modeled as squares indexed on 1 at the lowest bound and n at the highest bound. The second line specifies what happens at the first rank; providing a base case. The third line, the recursion, is the important part. It represents the A,B,C,D terms in the example. From this definition we can derive straightforward recursive code for q(i, j). In the following pseudocode, n is the size of the board, c(i, j) is the cost function, and min() returns the minimum of a number of values:

function minCost(i, j)
    if j < 1 or j > n
        return infinity
    else if i = 1
        return c(i, j)
    else
        return min( minCost(i-1, j-1), minCost(i-1, j), minCost(i-1, j+1) ) + c(i, j)

This function only computes the path cost, not the actual path. We discuss the actual path below. This, like the Fibonacci-numbers example, is horribly slow because it too exhibits the overlapping sub-problems attribute. That is, it recomputes the same path costs over and over. However, we can compute it much faster in a bottom-up fashion if we store path costs in a two-dimensional array q[i, j] rather than using a function. This avoids recomputation; all the values needed for array q[i, j] are computed ahead of time only once. Precomputed values for (i,j) are simply looked up whenever needed.

We also need to know what the actual shortest path is. To do this, we use another array p[i, j]; a predecessor array. This array records the path to any square s. The predecessor of s is modeled as an offset relative to the index (in q[i, j]) of the precomputed path cost of s. To reconstruct the complete path, we lookup the predecessor of s, then the predecessor of that square, then the predecessor of that square, and so on recursively, until we reach the starting square. Consider the following pseudocode:

function computeShortestPathArrays()
    for x from 1 to n
        q[1, x] := c(1, x)
    for y from 1 to n
        q[y, 0]     := infinity
        q[y, n + 1] := infinity
    for y from 2 to n
        for x from 1 to n
            m := min(q[y-1, x-1], q[y-1, x], q[y-1, x+1])
            q[y, x] := m + c(y, x)
            if m = q[y-1, x-1]
                p[y, x] := -1
            else if m = q[y-1, x]
                p[y, x] :=  0
            else
                p[y, x] :=  1

Now the rest is a simple matter of finding the minimum and printing it.

function computeShortestPath()
    computeShortestPathArrays()
    minIndex := 1
    min := q[n, 1]
    for i from 2 to n
        if q[n, i] < min
            minIndex := i
            min := q[n, i]
    printPath(n, minIndex)

function printPath(y, x)
    print(x)
    print("<-")
    if y = 2
        print(x + p[y, x])
    else
        printPath(y-1, x + p[y, x])

Sequence alignment

In genetics, sequence alignment is an important application where dynamic programming is essential. Typically, the problem consists of transforming one sequence into another using edit operations that replace, insert, or remove an element. Each operation has an associated cost, and the goal is to find the sequence of edits with the lowest total cost.

The problem can be stated naturally as a recursion, a sequence A is optimally edited into a sequence B by either:

1. inserting the first character of B, and performing an optimal alignment of A and the tail of B
2. deleting the first character of A, and performing the optimal alignment of the tail of A and B
3. replacing the first character of A with the first character of B, and performing optimal alignments of the tails of A and B.

The partial alignments can be tabulated in a matrix, where cell (i,j) contains the cost of the optimal alignment of A[1..i] to B[1..j]. The cost in cell (i,j) can be calculated by adding the cost of the relevant operations to the cost of its neighboring cells, and selecting the optimum.

Different variants exist, see Smith–Waterman algorithm and Needleman–Wunsch algorithm.

Tower of Hanoi puzzle

A model set of the Towers of Hanoi (with 8 disks)

 

An animated solution of the Tower of Hanoi puzzle for T(4,3).

The Tower of Hanoi or Towers of Hanoi is a mathematical game or puzzle. It consists of three rods, and a number of disks of different sizes which can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape.

The objective of the puzzle is to move the entire stack to another rod, obeying the following rules:

• Only one disk may be moved at a time.
• Each move consists of taking the upper disk from one of the rods and sliding it onto another rod, on top of the other disks that may already be present on that rod.
• No disk may be placed on top of a smaller disk.

The dynamic programming solution consists of solving the functional equation

S(n,h,t) = S(n-1,h, not(h,t)) ; S(1,h,t) ; S(n-1,not(h,t),t)

where n denotes the number of disks to be moved, h denotes the home rod, t denotes the target rod, not(h,t) denotes the third rod (neither h nor t), ";" denotes concatenation, and

S(n, h, t) := solution to a problem consisting of n disks that are to be moved from rod h to rod t.

For n=1 the problem is trivial, namely S(1,h,t) = "move a disk from rod h to rod t" (there is only one disk left).

The number of moves required by this solution is 2ⁿ − 1. If the objective is to maximize the number of moves (without cycling) then the dynamic programming functional equation is slightly more complicated and 3ⁿ − 1 moves are required.

Egg dropping puzzle

The following is a description of the instance of this famous puzzle involving N=2 eggs and a building with H=36 floors:

Suppose that we wish to know which stories in a 36-story building are safe to drop eggs from, and which will cause the eggs to break on landing (using U.S. English terminology, in which the first floor is at ground level). We make a few assumptions:

• An egg that survives a fall can be used again.
• A broken egg must be discarded.
• The effect of a fall is the same for all eggs.
• If an egg breaks when dropped, then it would break if dropped from a higher window.
• If an egg survives a fall, then it would survive a shorter fall.
• It is not ruled out that the first-floor windows break eggs, nor is it ruled out that eggs can survive the 36th-floor windows.

If only one egg is available and we wish to be sure of obtaining the right result, the experiment can be carried out in only one way. Drop the egg from the first-floor window; if it survives, drop it from the second-floor window. Continue upward until it breaks. In the worst case, this method may require 36 droppings. Suppose 2 eggs are available. What is the lowest number of egg-droppings that is guaranteed to work in all cases?

To derive a dynamic programming functional equation for this puzzle, let the state of the dynamic programming model be a pair s = (n,k), where

n = number of test eggs available, n = 0, 1, 2, 3, ..., N − 1.

k = number of (consecutive) floors yet to be tested, k = 0, 1, 2, ..., H − 1.

For instance, s = (2,6) indicates that two test eggs are available and 6 (consecutive) floors are yet to be tested. The initial state of the process is s = (N,H) where N denotes the number of test eggs available at the commencement of the experiment. The process terminates either when there are no more test eggs (n = 0) or when k = 0, whichever occurs first. If termination occurs at state s = (0,k) and k > 0, then the test failed.

Now, let

W(n,k) = minimum number of trials required to identify the value of the critical floor under the worst-case scenario given that the process is in state s = (n,k).

Then it can be shown that

W(n,k) = 1 + min{max(W(n − 1, x − 1), W(n,k − x)): x = 1, 2, ..., k }

with W(n,0) = 0 for all n > 0 and W(1,k) = k for all k. It is easy to solve this equation iteratively by systematically increasing the values of n and k.

Faster DP solution using a different parametrization

Notice that the above solution takes ${\displaystyle O(n{k}^2)}$ time with a DP solution. This can be improved to ${\displaystyle O(nk\log k)}$ time by binary searching on the optimal ${\displaystyle x}$ in the above recurrence, since ${\displaystyle W(n-1,x-1)}$ is increasing in ${\displaystyle x}$ while ${\displaystyle W(n,k-x)}$ is decreasing in ${\displaystyle x}$, thus a local minimum of ${\displaystyle max(W(n-1,x-1),W(n,k-x))}$ is a global minimum. Also, by storing the optimal ${\displaystyle x}$ for each cell in the DP table and referring to its value for the previous cell, the optimal ${\displaystyle x}$ for each cell can be found in constant time, improving it to ${\displaystyle O(nk)}$ time. However, there is an even faster solution that involves a different parametrization of the problem:

Let ${\displaystyle k}$ be the total number of floors such that the eggs break when dropped from the ${\displaystyle k}$th floor (The example above is equivalent to taking ${\displaystyle k=37}$).

Let ${\displaystyle m}$ be the minimum floor from which the egg must be dropped to be broken.

Let ${\displaystyle f(t,n)}$ be the maximum number of values of ${\displaystyle m}$ that are distinguishable using ${\displaystyle t}$ tries and ${\displaystyle n}$ eggs.

Then ${\displaystyle f(t,0)=f(0,n)=1}$ for all ${\displaystyle t,n\ge 0}$.

Let ${\displaystyle a}$ be the floor from which the first egg is dropped in the optimal strategy.

If the first egg broke, ${\displaystyle m}$ is from ${\displaystyle 1}$ to ${\displaystyle a}$ and distinguishable using at most ${\displaystyle t-1}$ tries and ${\displaystyle n-1}$ eggs.

If the first egg did not break, ${\displaystyle m}$ is from ${\displaystyle a+1}$ to ${\displaystyle k}$ and distinguishable using ${\displaystyle t-1}$ tries and ${\displaystyle n}$ eggs.

Therefore, ${\displaystyle f(t,n)=f(t-1,n-1)+f(t-1,n)}$.

Then the problem is equivalent to finding the minimum ${\displaystyle x}$ such that ${\displaystyle f(x,n)\ge k}$.

To do so, we could compute ${\displaystyle \{f(t,i):0\le i\le n\}}$ in order of increasing ${\displaystyle t}$, which would take ${\displaystyle O(nx)}$ time.

Thus, if we separately handle the case of ${\displaystyle n=1}$, the algorithm would take ${\displaystyle O(n\sqrt{k})}$ time.

But the recurrence relation can in fact be solved, giving ${\displaystyle f(t,n)=\sum _{i=0}^n(\genfrac{}{}{0ex}{}{t}{i})}$, which can be computed in ${\displaystyle O(n)}$ time using the identity ${\displaystyle (\genfrac{}{}{0ex}{}{t}{i+1})=(\genfrac{}{}{0ex}{}{t}{i})\frac{t-i}{i+1}}$ for all ${\displaystyle i\ge 0}$.

Since ${\displaystyle f(t,n)\le f(t+1,n)}$ for all ${\displaystyle t\ge 0}$, we can binary search on ${\displaystyle t}$ to find ${\displaystyle x}$, giving an ${\displaystyle O(n\log k)}$ algorithm.

Matrix chain multiplication

Main article: Matrix chain multiplication

Matrix chain multiplication is a well-known example that demonstrates utility of dynamic programming. For example, engineering applications often have to multiply a chain of matrices. It is not surprising to find matrices of large dimensions, for example 100×100. Therefore, our task is to multiply matrices ${\displaystyle A_1,A_2,....A_n}$. Matrix multiplication is not commutative, but is associative; and we can multiply only two matrices at a time. So, we can multiply this chain of matrices in many different ways, for example:

((A₁ × A₂) × A₃) × ... A_n

A₁×(((A₂×A₃)× ... ) × A_n)

(A₁ × A₂) × (A₃ × ... A_n)

and so on. There are numerous ways to multiply this chain of matrices. They will all produce the same final result, however they will take more or less time to compute, based on which particular matrices are multiplied. If matrix A has dimensions m×n and matrix B has dimensions n×q, then matrix C=A×B will have dimensions m×q, and will require m*n*q scalar multiplications (using a simplistic matrix multiplication algorithm for purposes of illustration).

For example, let us multiply matrices A, B and C. Let us assume that their dimensions are m×n, n×p, and p×s, respectively. Matrix A×B×C will be of size m×s and can be calculated in two ways shown below:

1. Ax(B×C) This order of matrix multiplication will require nps + mns scalar multiplications.
2. (A×B)×C This order of matrix multiplication will require mnp + mps scalar calculations.

Let us assume that m = 10, n = 100, p = 10 and s = 1000. So, the first way to multiply the chain will require 1,000,000 + 1,000,000 calculations. The second way will require only 10,000+100,000 calculations. Obviously, the second way is faster, and we should multiply the matrices using that arrangement of parenthesis.

Therefore, our conclusion is that the order of parenthesis matters, and that our task is to find the optimal order of parenthesis.

At this point, we have several choices, one of which is to design a dynamic programming algorithm that will split the problem into overlapping problems and calculate the optimal arrangement of parenthesis. The dynamic programming solution is presented below.

Let's call m[i,j] the minimum number of scalar multiplications needed to multiply a chain of matrices from matrix i to matrix j (i.e. A_i × .... × A_j, i.e. i<=j). We split the chain at some matrix k, such that i <= k < j, and try to find out which combination produces minimum m[i,j].

The formula is:

       if i = j, m[i,j]= 0
       if i < j, m[i,j]= min over all possible values of k (m[i,k]+m[k+1,j] + ${\displaystyle p_{i-1}\ast p_k\ast p_j}$) 

where k ranges from i to j − 1.

• ${\displaystyle p_{i-1}}$ is the row dimension of matrix i,
• ${\displaystyle p_k}$ is the column dimension of matrix k,
• ${\displaystyle p_j}$ is the column dimension of matrix j.

This formula can be coded as shown below, where input parameter "chain" is the chain of matrices, i.e. ${\displaystyle A_1,A_2,...A_n}$:

function OptimalMatrixChainParenthesis(chain)
    n = length(chain)
    for i = 1, n
        m[i,i] = 0    // Since it takes no calculations to multiply one matrix
    for len = 2, n
        for i = 1, n - len + 1
            j = i + len -1
            m[i,j] = infinity      // So that the first calculation updates
            for k = i, j-1
                q = m[i, k] + m[k+1, j] + ${\displaystyle p_{i-1}\ast p_k\ast p_j}$
                if q < m[i, j]     // The new order of parentheses is better than what we had
                    m[i, j] = q    // Update
                    s[i, j] = k    // Record which k to split on, i.e. where to place the parenthesis

So far, we have calculated values for all possible m[i, j], the minimum number of calculations to multiply a chain from matrix i to matrix j, and we have recorded the corresponding "split point"s[i, j]. For example, if we are multiplying chain A₁×A₂×A₃×A₄, and it turns out that m[1, 3] = 100 and s[1, 3] = 2, that means that the optimal placement of parenthesis for matrices 1 to 3 is ${\displaystyle (A_1\times A_2)\times A_3}$ and to multiply those matrices will require 100 scalar calculations.

This algorithm will produce "tables" m[, ] and s[, ] that will have entries for all possible values of i and j. The final solution for the entire chain is m[1, n], with corresponding split at s[1, n]. Unraveling the solution will be recursive, starting from the top and continuing until we reach the base case, i.e. multiplication of single matrices.

Therefore, the next step is to actually split the chain, i.e. to place the parenthesis where they (optimally) belong. For this purpose we could use the following algorithm:

function PrintOptimalParenthesis(s, i, j)
    if i = j
        print "A"i
    else
        print "(" 
        PrintOptimalParenthesis(s, i, s[i, j]) 
        PrintOptimalParenthesis(s, s[i, j] + 1, j) 
        print ")"

Of course, this algorithm is not useful for actual multiplication. This algorithm is just a user-friendly way to see what the result looks like.

To actually multiply the matrices using the proper splits, we need the following algorithm:

   function MatrixChainMultiply(chain from 1 to n)       // returns the final matrix, i.e. A1×A2×... ×An
      OptimalMatrixChainParenthesis(chain from 1 to n)   // this will produce s[ . ] and m[ . ] "tables"
      OptimalMatrixMultiplication(s, chain from 1 to n)  // actually multiply

   function OptimalMatrixMultiplication(s, i, j)   // returns the result of multiplying a chain of matrices from Ai to Aj in optimal way
      if i < j
         // keep on splitting the chain and multiplying the matrices in left and right sides
         LeftSide = OptimalMatrixMultiplication(s, i, s[i, j])
         RightSide = OptimalMatrixMultiplication(s, s[i, j] + 1, j)
         return MatrixMultiply(LeftSide, RightSide) 
      else if i = j
         return Ai   // matrix at position i
      else 
         print "error, i <= j must hold"

    function MatrixMultiply(A, B)    // function that multiplies two matrices
      if columns(A) = rows(B) 
         for i = 1, rows(A)
            for j = 1, columns(B)
               C[i, j] = 0
               for k = 1, columns(A)
                   C[i, j] = C[i, j] + A[i, k]*B[k, j] 
               return C 
      else 
          print "error, incompatible dimensions."

History

The term dynamic programming was originally used in the 1940s by Richard Bellman to describe the process of solving problems where one needs to find the best decisions one after another. By 1953, he refined this to the modern meaning, referring specifically to nesting smaller decision problems inside larger decisions, and the field was thereafter recognized by the IEEE as a systems analysis and engineering topic. Bellman's contribution is remembered in the name of the Bellman equation, a central result of dynamic programming which restates an optimization problem in recursive form.

Bellman explains the reasoning behind the term dynamic programming in his autobiography, Eye of the Hurricane: An Autobiography:

I spent the Fall quarter (of 1950) at RAND. My first task was to find a name for multistage decision processes. An interesting question is, "Where did the name, dynamic programming, come from?" The 1950s were not good years for mathematical research. We had a very interesting gentleman in Washington named Wilson. He was Secretary of Defense, and he actually had a pathological fear and hatred of the word "research". I’m not using the term lightly; I’m using it precisely. His face would suffuse, he would turn red, and he would get violent if people used the term research in his presence. You can imagine how he felt, then, about the term mathematical. The RAND Corporation was employed by the Air Force, and the Air Force had Wilson as its boss, essentially. Hence, I felt I had to do something to shield Wilson and the Air Force from the fact that I was really doing mathematics inside the RAND Corporation. What title, what name, could I choose? In the first place I was interested in planning, in decision making, in thinking. But planning, is not a good word for various reasons. I decided therefore to use the word "programming". I wanted to get across the idea that this was dynamic, this was multistage, this was time-varying. I thought, let's kill two birds with one stone. Let's take a word that has an absolutely precise meaning, namely dynamic, in the classical physical sense. It also has a very interesting property as an adjective, and that is it's impossible to use the word dynamic in a pejorative sense. Try thinking of some combination that will possibly give it a pejorative meaning. It's impossible. Thus, I thought dynamic programming was a good name. It was something not even a Congressman could object to. So I used it as an umbrella for my activities.

— Richard Bellman, Eye of the Hurricane: An Autobiography (1984, page 159)

The word dynamic was chosen by Bellman to capture the time-varying aspect of the problems, and because it sounded impressive. The word programming referred to the use of the method to find an optimal program, in the sense of a military schedule for training or logistics. This usage is the same as that in the phrases linear programming and mathematical programming, a synonym for mathematical optimization.

The above explanation of the origin of the term is lacking. As Russell and Norvig in their book have written, referring to the above story: "This cannot be strictly true, because his first paper using the term (Bellman, 1952) appeared before Wilson became Secretary of Defense in 1953." Also, there is a comment in a speech by Harold J. Kushner, where he remembers Bellman. Quoting Kushner as he speaks of Bellman: "On the other hand, when I asked him the same question, he replied that he was trying to upstage Dantzig's linear programming by adding dynamic. Perhaps both motivations were true."

Algorithms that use dynamic programming

• Recurrent solutions to lattice models for protein-DNA binding
• Backward induction as a solution method for finite-horizon discrete-time dynamic optimization problems
• Method of undetermined coefficients can be used to solve the Bellman equation in infinite-horizon, discrete-time, discounted, time-invariant dynamic optimization problems
• Many string algorithms including longest common subsequence, longest increasing subsequence, longest common substring, Levenshtein distance (edit distance)
• Many algorithmic problems on graphs can be solved efficiently for graphs of bounded treewidth or bounded clique-width by using dynamic programming on a tree decomposition of the graph.
• The Cocke–Younger–Kasami (CYK) algorithm which determines whether and how a given string can be generated by a given context-free grammar
• Knuth's word wrapping algorithm that minimizes raggedness when word wrapping text
• The use of transposition tables and refutation tables in computer chess
• The Viterbi algorithm (used for hidden Markov models, and particularly in part of speech tagging)
• The Earley algorithm (a type of chart parser)
• The Needleman–Wunsch algorithm and other algorithms used in bioinformatics, including sequence alignment, structural alignment, RNA structure prediction
• Floyd's all-pairs shortest path algorithm
• Optimizing the order for chain matrix multiplication
• Pseudo-polynomial time algorithms for the subset sum, knapsack and partition problems
• The dynamic time warping algorithm for computing the global distance between two time series
• The Selinger (a.k.a. System R) algorithm for relational database query optimization
• De Boor algorithm for evaluating B-spline curves
• Duckworth–Lewis method for resolving the problem when games of cricket are interrupted
• The value iteration method for solving Markov decision processes
• Some graphic image edge following selection methods such as the "magnet" selection tool in Photoshop
• Some methods for solving interval scheduling problems
• Some methods for solving the travelling salesman problem, either exactly (in exponential time) or approximately (e.g. via the bitonic tour)
• Recursive least squares method
• Beat tracking in music information retrieval
• Adaptive-critic training strategy for artificial neural networks
• Stereo algorithms for solving the correspondence problem used in stereo vision
• Seam carving (content-aware image resizing)
• The Bellman–Ford algorithm for finding the shortest distance in a graph
• Some approximate solution methods for the linear search problem
• Kadane's algorithm for the maximum subarray problem
• Optimization of electric generation expansion plans in the Wein Automatic System Planning (WASP) package

Alcohol-related dementia

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Alcohol-related_dementia 

Alcohol-related dementia (ARD) is a form of dementia caused by long-term, excessive consumption of alcoholic beverages, resulting in neurological damage and impaired cognitive function.

Terminology

Alcohol-related dementia is a broad term currently preferred among medical professionals. If a person has alcohol-related ‘dementia’ they will struggle with day-to-day tasks. This is because of the damage to their brain, caused by regularly drinking too much alcohol over many years.  This affects memory, learning and other mental functions. Korsakoff’s syndrome and Wernicke-Korsakoff syndrome are particular forms of alcohol related brain injury which may be related to alcohol related dementia. Many experts use the terms alcohol (or alcoholic) dementia to describe a specific form of ARD, characterized by impaired executive function (planning, thinking, and judgment). Another form of ARD is known as wet brain (Wernicke–Korsakoff syndrome), characterized by short-term memory loss and thiamine (vitamin B1) deficiency. ARD patients often have symptoms of both forms, i.e. impaired ability to plan, apathy, and memory loss. ARD may occur with other forms of dementia (mixed dementia). The diagnosis of ARD is widely recognized but rarely applied, due to a lack of specific diagnostic criteria.

On many non-medical websites, the terms wet brain and alcohol-related dementia are often used interchangeably, creating significant confusion. Additionally, the term alcohol-induced persistent dementia is another nonspecific name that is sometimes used.

Signs and symptoms

Alcohol-related dementia presents as a global deterioration in intellectual function with memory not being specifically affected, but it may occur with other forms of dementia, resulting in a wide range of symptoms. Certain individuals with alcohol-related dementia present with damage to the frontal lobes of their brain causing disinhibition, loss of planning and executive functions, and a disregard for the consequences of their behavior. Other types of alcohol-related dementia such as Wernicke encephalopathy cause the destruction of certain areas of the brain, where changes in memory, primarily a loss of short-term memory, are the main symptom. Most presentations of alcohol dementia are somewhere along the spectrum between a global dementia and Korsakoff's psychosis, and may include symptoms of both.

Individuals affected by alcohol-related dementia may develop memory problems, language impairment, and an inability to perform complex motor tasks such as getting dressed. Heavy alcohol consumption also damages the nerves in arms and legs, i.e. peripheral neuropathy, as well as the cerebellum that controls coordination thereby leading to the development of cerebellar ataxia. These patients frequently have problems with sensation in their extremities and may demonstrate unsteadiness on their feet.

Alcohol-related dementia can produce a variety of psychiatric problems including psychosis (disconnection from reality), depression, anxiety, and personality changes. Patients with alcoholic dementia often develop apathy, related to frontal lobe damage, that may mimic depression. People with an alcohol use disorder are more likely to become depressed than people without alcohol use disorder, and it may be difficult to differentiate between depression and alcohol dementia.

Pathophysiology

Epidemiological studies show an association between long-term alcohol intoxication and dementia. Alcohol can damage the brain directly as a neurotoxin, or it can damage it indirectly by causing malnutrition, primarily a loss of thiamine (vitamin B1). Alcohol use disorder is common in older persons, and alcohol-related dementia is under-diagnosed.

Diagnosis

The signs and symptoms of alcohol-related dementia are essentially the same as the symptoms present in other types of dementia, making alcohol-related dementia difficult to diagnose. There are very few qualitative differences between alcohol dementia and Alzheimer's disease and it is therefore difficult to distinguish between the two. Some of these warning signs may include memory loss, difficulty performing familiar tasks, poor or impaired judgment and problems with language. However the biggest indicator is friends or family members reporting changes in personality.

A simple test for intellectual function, like the Folstein Mini-Mental Status Examination, is the minimum screen for dementia. The test requires 15–20 minutes to administer and is available in mental health centers.

Diagnosing alcohol-related dementia can be difficult due to the wide range of symptoms and a lack of specific brain pathology. The Diagnostic and Statistical Manual of Mental Disorders (DSM-IV) is a guide to aid doctors in diagnosing a range of psychiatric disorders, and may be helpful in diagnosing dementia.

Diagnostic criteria

The existence of alcohol-related dementia is widely acknowledged but not often used as a diagnosis, due to a lack of widely accepted, non-subjective diagnostic criteria; more research is needed. Criteria for alcohol-induced persistent dementia in the Diagnostic and Statistical Manual of Mental Disorders (DSM-IV) include the following:

A. The development of multiple cognitive deficits manifested by both:

1. Memory impairment (impaired ability to learn new information or to recall previously learned information)
2. One (or more) of the following cognitive disturbances:

• (a) Aphasia (language disturbance)
• (b) Apraxia (impaired ability to carry out motor activities despite intact motor function)
• (c) Agnosia (failure to recognize or identify objects despite intact sensory function)
• (d) Disturbance in executive functioning (i.e. planning, organizing, sequencing, abstracting)

B. The cognitive deficits in criteria A1 and A2 each cause significant impairment in social or occupational functioning and represent a significant decline from a previous level of functioning.

C. The deficits do not occur exclusively during the course of a delirium and persist beyond the usual duration of substance intoxication or withdrawal.

D. There is evidence from the history, physical examination, or laboratory findings that deficits are etiologically related to the persisting effects of substance use.

There are problems with DSM diagnostic criteria. First, they are vague and subjective. Furthermore, the criteria for diagnosis of dementia were inspired by the clinical presentation of Alzheimer's disease and are poorly adapted to the diagnosis of other dementias. This has led to efforts to develop better diagnostic models.

Oslin (Int J Geriatr Psychiatry 1998) proposed alternative clinical diagnostic criteria which were validated. The criteria include a clinical diagnosis of dementia at least 60 days after last exposure to alcohol, significant alcohol use (i.e. minimum 35 standard drinks/week for males and 28 for women) for more than five years, and significant alcohol use occurring within three years of the initial onset of cognitive deficits. Oslin proposed the new and refined diagnostic criteria for alcohol-related dementia because he hoped that the redefined classification system would bring more awareness and clarity to the relationship between alcohol use and dementia.

Oslin's proposed classification of ARD:

• Definite alcohol-related dementia

At the current time there are no acceptable criteria to definitively define alcohol-related dementia.

• Probable alcohol-related dementia

A. The criteria for the clinical diagnosis of probable alcohol-related dementia include the following:

1. A clinical diagnosis of dementia at least 60 days after the last exposure to alcohol.
2. Significant alcohol use as defined by a minimum average of 35 standard drinks per week for men (28 for women) for greater than a period of five years. The period of significant alcohol use must occur within three years of the initial onset of dementia.

B. The diagnosis of alcohol-related dementia is supported by the presence of any of the following

1. Alcohol related hepatic, pancreatic, gastrointestinal, cardiovascular, or renal disease i.e. other end-organ damage.
2. Ataxia or peripheral sensory polyneuropathy (not attributed to other causes).
3. Beyond 60 days of abstinence, the cognitive impairment stabilizes or improves.
4. After 60 days of abstinence, any neuroimaging evidence of ventricular or sulcal dilatation improves.
5. Neuroimaging evidence of cerebellar atrophy, especially in the vermis.

C. The following clinical features cast doubt on the diagnosis of alcohol-related dementia

1. The presence of language impairment, especially dysnomia or anomia.
2. the presence of focal neurologic signs or symptoms (except ataxia or peripheral sensory polyneuropathy).
3. Neuroimaging evidence for cortical or subcortical infarction, subdural hematoma, or other focal brain pathology.
4. Elevated Hachinski Ischemia Scale score.

D. Clinical features that are neither supportive nor cast doubt on the diagnosis of alcohol-related dementia included:

1. Neuroimaging evidence of cortical atrophy.
2. The presence of periventricular or deep white matter lesions on neuroimaging in the absence of focal infarct(s).
3. The presence of the Apolipoprotein c4 allele.

Treatment

ARD is treated with abstinence from further alcohol consumption.

Prognosis

Multiple withdrawals and binge drinking may significantly exacerbate cognitive deficits. Older individuals are at greater risk of cognitive changes.

Recovery

Following abstinence, many deficits often resolve rapidly (in as little as a week). Further gradual recovery of cognitive abilities may take place over several years. Executive function, working memory, perceptual impairment, and motor impairments often persist after short-term abstinence. Recovery of cognitive skills appears correlated to recent intake levels and duration of abstinence, rather than to lifetime cumulative alcohol intake.

Older individuals are less likely to recover completely following cessation of alcohol intake.

Epidemiology

The onset of alcohol dementia can occur as early as age 30, although it is far more common that the dementia will reveal itself anywhere from age 50 to 70. The onset and the severity of this type of dementia is directly correlated to the amount of alcohol that a person consumes over their lifetime.

Sex appears to be a risk factor for cognitive impairment, with females being more susceptible despite lower alcohol intake.

A French study, looking at other studies of thousands of subjects, found that moderate alcohol consumption (up to four glasses of wine per week) was associated with lower levels of dementia, and vice versa. There is insufficient evidence to assume that alcohol is protective against dementia at any level of intake; some studies found the opposite effect, and the quality of evidence from current epidemiological studies is poor overall (since observational studies assessing health effects of alcohol intake cannot adequately control for confounding factors).

Notable cases

According to her family, the socialite Leonore Lemmon (fiancée of George Reeves) spent the last few years of her life with alcohol dementia, before dying in 1989.

The Australian entertainer and "King of Comedy" Graham Kennedy had alcohol-related dementia at time of his death in 2005.

Neurodegenerative disease

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Neurodegenerative_disease 

Neurodegenerative disease
Normal brain on left contrasted with structural changes shown in brain on right of person with Alzheimer's disease, the most common neurodegenerative disease
Specialty	Neurology, Psychiatry

A neurodegenerative disease is caused by the progressive loss of structure or function of neurons, in the process known as neurodegeneration. Such neuronal damage may ultimately involve cell death. Neurodegenerative diseases include amyotrophic lateral sclerosis, multiple sclerosis, Parkinson's disease, Alzheimer's disease, Huntington's disease, multiple system atrophy, and prion diseases. Neurodegeneration can be found in the brain at many different levels of neuronal circuitry, ranging from molecular to systemic. Because there is no known way to reverse the progressive degeneration of neurons, these diseases are considered to be incurable; however research has shown that the two major contributing factors to neurodegeneration are oxidative stress and inflammation. Biomedical research has revealed many similarities between these diseases at the subcellular level, including atypical protein assemblies (like proteinopathy) and induced cell death. These similarities suggest that therapeutic advances against one neurodegenerative disease might ameliorate other diseases as well.

Within neurodegenerative diseases, it is estimated that 55 million people worldwide had dementia in 2019, and that by 2050 this figure will increase to 139 million people.

Specific disorders

Alzheimer's disease

Comparison of brain tissue between healthy individual and Alzheimer's disease patient, demonstrating extent of neuronal death

 

Main article: Alzheimer's disease

Alzheimer's disease (AD) is a chronic neurodegenerative disease that results in the loss of neurons and synapses in the cerebral cortex and certain subcortical structures, resulting in gross atrophy of the temporal lobe, parietal lobe, and parts of the frontal cortex and cingulate gyrus. It is the most common neurodegenerative disease. Even with billions of dollars being used to find a treatment for Alzheimer's disease, no effective treatments have been found. However, clinical trials have developed certain compounds that could potentially change the future of Alzheimer's disease treatments. Currently, diagnoses of Alzheimer's is subpar, and better methods need to be utilized for various aspects of clinical diagnoses. Alzheimer's has a 20% misdiagnosis rate.

AD pathology is primarily characterized by the presence of amyloid plaques and neurofibrillary tangles. Plaques are made up of small peptides, typically 39–43 amino acids in length, called amyloid beta (also written as A-beta or Aβ). Amyloid beta is a fragment from a larger protein called amyloid precursor protein (APP), a transmembrane protein that penetrates through the neuron's membrane. APP appears to play roles in normal neuron growth, survival and post-injury repair. APP is cleaved into smaller fragments by enzymes such as gamma secretase and beta secretase. One of these fragments gives rise to fibrils of amyloid beta which can self-assemble into the dense extracellular amyloid plaques.

Parkinson's disease

Main article: Parkinson's disease

Parkinson's disease (PD) is the second most common neurodegenerative disorder. It typically manifests as bradykinesia, rigidity, resting tremor and posture instability. The crude prevalence rate of PD has been reported to range from 15 per 100,000 to 12,500 per 100,000, and the incidence of PD from 15 per 100,000 to 328 per 100,000, with the disease being less common in Asian countries.

PD is primarily characterized by death of dopaminergic neurons in the substantia nigra, a region of the midbrain. The cause of this selective cell death is unknown. Notably, alpha-synuclein-ubiquitin complexes and aggregates are observed to accumulate in Lewy bodies within affected neurons. It is thought that defects in protein transport machinery and regulation, such as RAB1, may play a role in this disease mechanism. Impaired axonal transport of alpha-synuclein may also lead to its accumulation in Lewy bodies. Experiments have revealed reduced transport rates of both wild-type and two familial Parkinson's disease-associated mutant alpha-synucleins through axons of cultured neurons. Membrane damage by alpha-synuclein could be another Parkinson's disease mechanism.

The main known risk factor is age. Mutations in genes such as α-synuclein (SNCA), leucine-rich repeat kinase 2 (LRRK2), glucocerebrosidase (GBA), and tau protein (MAPT) can also cause hereditary PD or increase PD risk. While PD is the second most common neurodegenerative disorder, problems with diagnoses still persist. Problems with the sense of smell is a widespread symptom of Parkinson's disease (PD), however, some neurologists question its efficacy. This assessment method is a source of controversy among medical professionals. The gut microbiome might play a role in the diagnosis of PD, and research suggests various ways that could revolutionize the future of PD treatment.

Huntington's disease

Main article: Huntington's disease

Huntington's disease (HD) is a rare autosomal dominant neurodegenerative disorder caused by mutations in the huntingtin gene (HTT). HD is characterized by loss of medium spiny neurons and astrogliosis. The first brain region to be substantially affected is the striatum, followed by degeneration of the frontal and temporal cortices. The striatum's subthalamic nuclei send control signals to the globus pallidus, which initiates and modulates motion. The weaker signals from subthalamic nuclei thus cause reduced initiation and modulation of movement, resulting in the characteristic movements of the disorder, notably chorea. Huntington's disease presents itself later in life even though the proteins that cause the disease works towards manifestation from their early stages in the humans affected by the proteins. Along with being a neurodegenerative disorder, HD has links to problems with neurodevelopment.

HD is caused by polyglutamine tract expansion in the huntingtin gene, resulting in the mutant huntingtin. Aggregates of mutant huntingtin form as inclusion bodies in neurons, and may be directly toxic. Additionally, they may damage molecular motors and microtubules to interfere with normal axonal transport, leading to impaired transport of important cargoes such as BDNF. Huntington's disease currently has no effective treatments that would modify the disease.

Multiple sclerosis

Main article: Multiple sclerosis

Multiple sclerosis (MS) is a chronic debilitating demyelinating disease of the central nervous system, caused by an autoimmune attack resulting in the progressive loss of myelin sheath on neuronal axons. The resultant decrease in the speed of signal transduction leads to a loss of functionality that includes both cognitive and motor impairment depending on the location of the lesion. The progression of MS occurs due to episodes of increasing inflammation, which is proposed to be due to the release of antigens such as myelin oligodendrocyte glycoprotein, myelin basic protein, and proteolipid protein, causing an autoimmune response. This sets off a cascade of signaling molecules that result in T cells, B cells, and Macrophages to cross the blood-brain barrier and attack myelin on neuronal axons leading to inflammation. Further release of antigens drives subsequent degeneration causing increased inflammation. Multiple sclerosis presents itself as a spectrum based on the degree of inflammation, a majority of patients experience early relapsing and remitting episodes of neuronal deterioration following a period of recovery. Some of these individuals may transition to a more linear progression of the disease, while about 15% of others begin with a progressive course on the onset of Multiple sclerosis. The inflammatory response contributes to the loss of the grey matter, and as a result current literature devotes itself to combatting the auto-inflammatory aspect of the disease. While there are several proposed causal links between EBV and the HLA-DRB1*15:01 allele to the onset of MS – they may contribute to the degree of autoimmune attack and the resultant inflammation – they do not determine the onset of MS.

Amyotrophic lateral sclerosis

Main article: Amyotrophic lateral sclerosis

Amyotrophic lateral sclerosis (ALS) or Lou Gehrig's disease is a disease in which motor neurons are selectively targeted for degeneration. Amyotrophic lateral sclerosis (ALS) is a neurodegenerative disorder that negatively impacts the upper motor neurons (UMNs) and lower motor neurons (LMNs). In 1993, missense mutations in the gene encoding the antioxidant enzyme Cu/Zn superoxide dismutase 1 (SOD1) were discovered in a subsets of patients with familial ALS. This discovery led researchers to focus on unlocking the mechanisms for SOD1-mediated diseases. However, the pathogenic mechanism underlying SOD1 mutant toxicity has yet to be resolved. More recently, TDP-43 and FUS protein aggregates have been implicated in some cases of the disease, and a mutation in chromosome 9 (C9orf72) is thought to be the most common known cause of sporadic ALS. It is diagnosed by skeletal muscle weakness that progresses gradually. Early diagnosis of ALS is harder than with other neurodegenerative diseases as there are no highly effective means of determining its early onset. Currently, there is research being done regarding the diagnosis of ALS through upper motor neuron tests. The Penn Upper Motor Neuron Score (PUMNS) consists of 28 criteria with a score range of 0-32. A higher score indicates a higher level of burden present on the upper motor neurons. The PUMNS has proven quite effective in determining the burden that exists on upper motor neurons in affected patients.

Independent research provided in vitro evidence that the primary cellular sites where SOD1 mutations act are located on astrocytes. Astrocytes then cause the toxic effects on the motor neurons. The specific mechanism of toxicity still needs to be investigated, but the findings are significant because they implicate cells other than neuron cells in neurodegeneration.

Batten disease

Main article: Batten disease

Batten disease is a rare and fatal recessive neurodegenerative disorder that begins in childhood. Batten disease is the common name for a group of lysosomal storage disorders known as neuronal ceroid lipofuscinoses (NCLs) – each caused by a specific gene mutation, of which there are thirteen. Since Batten disease is quite rare, its worldwide prevalence is about 1 in every 100,000 live births. In North America, CLN3 disease (juvenile NCL) typically manifests between the ages of 4 to 7. Batten disease is characterized by motor impairment, epilepsy, dementia, vision loss, and shortened lifespan. A loss of vision is common first sign of Batten disease. Loss of vision is typically preceded by cognitive and behavioral changes, seizures, and loss of the ability to walk. It is common for people to establish cardiac arrhythmias and difficulties eating food as the disease progresses. Batten disease diagnosis depends on a conflation of many criteria: clinical signs and symptoms, evaluations of the eye, electroencephalograms (EEG), and brain magnetic resonance imaging (MRI) results. The diagnosis provided by these results are corroborated by genetic and biochemical testing. No effective treatments were available to prevent the disease from being widespread before the past few years. In recent years, more models have been created to expedite the research process for methods to treat Batten disease.

Creutzfeldt–Jakob disease

Main article: Creutzfeldt-Jakob disease

Creutzfeldt–Jakob disease (CJD) is a prion disease that is characterized by rapidly progressive dementia. Abnormal proteins called prions aggregate in brain tissue leading to nerve cell death. Prions are misfolded PRNP proteins. They are also infectious. Variant Creutzfeldt–Jakob disease (vCJD) is the infectious form that comes from the meat of a cow that was infected with bovine spongiform encephalopathy, also called mad cow disease.

Risk factor

Further information: Aging brain

The greatest risk factor for neurodegenerative diseases is aging. Mitochondrial DNA mutations as well as oxidative stress both contribute to aging. Many of these diseases are late-onset, meaning there is some factor that changes as a person ages for each disease. One constant factor is that in each disease, neurons gradually lose function as the disease progresses with age. It has been proposed that DNA damage accumulation provides the underlying causative link between aging and neurodegenerative disease. About 20–40% of healthy people between 60 and 78 years old experience discernable decrements in cognitive performance in several domains including working, spatial, and episodic memory, and processing speed.

Mechanisms

Genetics

See also: Trinucleotide repeat disorder and Epigenetics of neurodegenerative diseases

Many neurodegenerative diseases are caused by genetic mutations, most of which are located in completely unrelated genes. In many of the different diseases, the mutated gene has a common feature: a repeat of the CAG nucleotide triplet. CAG codes for the amino acid glutamine. A repeat of CAG results in a polyglutamine (polyQ) tract. Diseases associated with such mutations are known as trinucleotide repeat disorders.

Polyglutamine repeats typically cause dominant pathogenesis. Extra glutamine residues can acquire toxic properties through a variety of ways, including irregular protein folding and degradation pathways, altered subcellular localization, and abnormal interactions with other cellular proteins. PolyQ studies often use a variety of animal models because there is such a clearly defined trigger – repeat expansion. Extensive research has been done using the models of nematode (C. elegans), and fruit fly (Drosophila), mice, and non-human primates.

Nine inherited neurodegenerative diseases are caused by the expansion of the CAG trinucleotide and polyQ tract, including Huntington's disease and the spinocerebellar ataxias.

Epigenetics

The presence of epigenetic modifications for certain genes has been demonstrated in this type of pathology. An example is FKBP5 gene, which progressively increases its expression with age and has been related to Braak staging and increased tau pathology both in vitro and in mouse models of AD.

Protein misfolding

See also: Stress granule

Several neurodegenerative diseases are classified as proteopathies as they are associated with the aggregation of misfolded proteins. Protein toxicity is one of the key mechanisms of many neurodegenrative diseases.

• alpha-synuclein: can aggregate to form insoluble fibrils in pathological conditions characterized by Lewy bodies, such as Parkinson's disease, dementia with Lewy bodies, and multiple system atrophy. Alpha-synuclein is the primary structural component of Lewy body fibrils. In addition, an alpha-synuclein fragment, known as the non-Abeta component (NAC), is found in amyloid plaques in Alzheimer's disease.
• tau: hyperphosphorylated tau protein is the main component of neurofibrillary tangles in Alzheimer's disease; tau fibrils are the main component of Pick bodies found in behavioral variant frontotemporal dementia.
• amyloid beta: the major component of amyloid plaques in Alzheimer's disease.
• prion: main component of prion diseases and transmissible spongiform encephalopathy.

Intracellular mechanisms

Protein degradation pathways

Parkinson's disease and Huntington's disease are both late-onset and associated with the accumulation of intracellular toxic proteins. Diseases caused by the aggregation of proteins are known as proteopathies, and they are primarily caused by aggregates in the following structures:ytosol, e.g. Parkinson's and Huntington's

• nucleus, e.g. Spinocerebellar ataxia type 1 (SCA1)
• endoplasmic reticulum (ER), (as seen with neuroserpin mutations that cause familial encephalopathy with neuroserpin inclusion bodies)
• extracellularly excreted proteins, amyloid-beta in Alzheimer's disease

There are two main avenues eukaryotic cells use to remove troublesome proteins or organelles:

• ubiquitin–proteasome: protein ubiquitin along with enzymes is key for the degradation of many proteins that cause proteopathies including polyQ expansions and alpha-synucleins. Research indicates proteasome enzymes may not be able to correctly cleave these irregular proteins, which could possibly result in a more toxic species. This is the primary route cells use to degrade proteins.
  • Decreased proteasome activity is consistent with models in which intracellular protein aggregates form. It is still unknown whether or not these aggregates are a cause or a result of neurodegeneration.
• autophagy–lysosome pathways: a form of programmed cell death (PCD), this becomes the favorable route when a protein is aggregate-prone meaning it is a poor proteasome substrate. This can be split into two forms of autophagy: macroautophagy and chaperone-mediated autophagy (CMA).
  • macroautophagy is involved with nutrient recycling of macromolecules under conditions of starvation, certain apoptotic pathways, and if absent, leads to the formation of ubiquinated inclusions. Experiments in mice with neuronally confined macroautophagy-gene knockouts develop intraneuronal aggregates leading to neurodegeneration.
  • chaperone-mediated autophagy defects may also lead to neurodegeneration. Research has shown that mutant proteins bind to the CMA-pathway receptors on lysosomal membrane and in doing so block their own degradation as well as the degradation of other substrates.

Membrane damage

Damage to the membranes of organelles by monomeric or oligomeric proteins could also contribute to these diseases. Alpha-synuclein can damage membranes by inducing membrane curvature, and cause extensive tubulation and vesiculation when incubated with artificial phospholipid vesicles. The tubes formed from these lipid vesicles consist of both micellar as well as bilayer tubes. Extensive induction of membrane curvature is deleterious to the cell and would eventually lead to cell death. Apart from tubular structures, alpha-synuclein can also form lipoprotein nanoparticles similar to apolipoproteins.

Mitochondrial dysfunction

The most common form of cell death in neurodegeneration is through the intrinsic mitochondrial apoptotic pathway. This pathway controls the activation of caspase-9 by regulating the release of cytochrome c from the mitochondrial intermembrane space. Reactive oxygen species (ROS) are normal byproducts of mitochondrial respiratory chain activity. ROS concentration is mediated by mitochondrial antioxidants such as manganese superoxide dismutase (SOD2) and glutathione peroxidase. Over production of ROS (oxidative stress) is a central feature of all neurodegenerative disorders. In addition to the generation of ROS, mitochondria are also involved with life-sustaining functions including calcium homeostasis, PCD, mitochondrial fission and fusion, lipid concentration of the mitochondrial membranes, and the mitochondrial permeability transition. Mitochondrial disease leading to neurodegeneration is likely, at least on some level, to involve all of these functions.

There is strong evidence that mitochondrial dysfunction and oxidative stress play a causal role in neurodegenerative disease pathogenesis, including in four of the more well known diseases Alzheimer's, Parkinson's, Huntington's, and amyotrophic lateral sclerosis.

Neurons are particularly vulnerable to oxidative damage due to their strong metabolic activity associated with high transcription levels, high oxygen consumption, and weak antioxidant defense.

DNA damage

The brain metabolizes as much as a fifth of consumed oxygen, and reactive oxygen species produced by oxidative metabolism are a major source of DNA damage in the brain. Damage to a cell's DNA is particularly harmful because DNA is the blueprint for protein production and unlike other molecules it cannot simply be replaced by re-synthesis. The vulnerability of post-mitotic neurons to DNA damage (such as oxidative lesions or certain types of DNA strand breaks), coupled with a gradual decline in the activities of repair mechanisms, could lead to accumulation of DNA damage with age and contribute to brain aging and neurodegeneration. DNA single-strand breaks are common and are associated with the neurodegenerative disease ataxia-oculomotor apraxia. Increased oxidative DNA damage in the brain is associated with Alzheimer's disease and Parkinson's disease. Defective DNA repair has been linked to neurodegenerative disorders such as Alzheimer's disease, amyotrophic lateral sclerosis, ataxia telangiectasia, Cockayne syndrome, Parkinson's disease and xeroderma pigmentosum.

Axonal transport

Main article: Axonal transport

Axonal swelling, and axonal spheroids have been observed in many different neurodegenerative diseases. This suggests that defective axons are not only present in diseased neurons, but also that they may cause certain pathological insult due to accumulation of organelles. Axonal transport can be disrupted by a variety of mechanisms including damage to: kinesin and cytoplasmic dynein, microtubules, cargoes, and mitochondria. When axonal transport is severely disrupted a degenerative pathway known as Wallerian-like degeneration is often triggered.

Programmed cell death

Programmed cell death (PCD) is death of a cell in any form, mediated by an intracellular program. This process can be activated in neurodegenerative diseases including Parkinson's disease, amytrophic lateral sclerosis, Alzheimer's disease and Huntington's disease. PCD observed in neurodegenerative diseases may be directly pathogenic; alternatively, PCD may occur in response to other injury or disease processes.

Apoptosis (type I)

Apoptosis is a form of programmed cell death in multicellular organisms. It is one of the main types of programmed cell death (PCD) and involves a series of biochemical events leading to a characteristic cell morphology and death.

• Extrinsic apoptotic pathways: Occur when factors outside the cell activate cell surface death receptors (e.g., Fas) that result in the activation of caspases-8 or -10.
• Intrinsic apoptotic pathways: Result from mitochondrial release of cytochrome c or endoplasmic reticulum malfunctions, each leading to the activation of caspase-9. The nucleus and Golgi apparatus are other organelles that have damage sensors, which can lead the cells down apoptotic pathways.

Caspases (cysteine-aspartic acid proteases) cleave at very specific amino acid residues. There are two types of caspases: initiators and effectors. Initiator caspases cleave inactive forms of effector caspases. This activates the effectors that in turn cleave other proteins resulting in apoptotic initiation.

Autophagic (type II)

Autophagy is a form of intracellular phagocytosis in which a cell actively consumes damaged organelles or misfolded proteins by encapsulating them into an autophagosome, which fuses with a lysosome to destroy the contents of the autophagosome. Because many neurodegenerative diseases show unusual protein aggregates, it is hypothesized that defects in autophagy could be a common mechanism of neurodegeneration.

Cytoplasmic (type III)

PCD can also occur via non-apoptotic processes, also known as Type III or cytoplasmic cell death. For example, type III PCD might be caused by trophotoxicity, or hyperactivation of trophic factor receptors. Cytotoxins that induce PCD can cause necrosis at low concentrations, or aponecrosis (combination of apoptosis and necrosis) at higher concentrations. It is still unclear exactly what combination of apoptosis, non-apoptosis, and necrosis causes different kinds of aponecrosis.

Transglutaminase

Transglutaminases are human enzymes ubiquitously present in the human body and in the brain in particular.

The main function of transglutaminases is bind proteins and peptides intra- and intermolecularly, by a type of covalent bonds termed isopeptide bonds, in a reaction termed transamidation or crosslinking.

Transglutaminase binding of these proteins and peptides make them clump together. The resulting structures are turned extremely resistant to chemical and mechanical disruption.

Most relevant human neurodegenerative diseases share the property of having abnormal structures made up of proteins and peptides.

Each of these neurodegenerative diseases have one (or several) specific main protein or peptide. In Alzheimer's disease, these are amyloid-beta and tau. In Parkinson's disease, it is alpha-synuclein. In Huntington's disease, it is huntingtin.

Transglutaminase substrates: Amyloid-beta, tau, alpha-synuclein and huntingtin have been proved to be substrates of transglutaminases in vitro or in vivo, that is, they can be bonded by trasglutaminases by covalent bonds to each other and potentially to any other transglutaminase substrate in the brain.

Transglutaminase augmented expression: It has been proved that in these neurodegenerative diseases (Alzheimer's disease, Parkinson's disease, and Huntington's disease) the expression of the transglutaminase enzyme is increased.

Presence of isopeptide bonds in these structures: The presence of isopeptide bonds (the result of the transglutaminase reaction) have been detected in the abnormal structures that are characteristic of these neurodegenerative diseases.

Co-localization: Co-localization of transglutaminase mediated isopeptide bonds with these abnormal structures has been detected in the autopsy of brains of patients with these diseases.

Management

The process of neurodegeneration is not well understood, so the diseases that stem from it have, as yet, no cures.

Animal models in research

In the search for effective treatments (as opposed to palliative care), investigators employ animal models of disease to test potential therapeutic agents. Model organisms provide an inexpensive and relatively quick means to perform two main functions: target identification and target validation. Together, these help show the value of any specific therapeutic strategies and drugs when attempting to ameliorate disease severity. An example is the drug Dimebon by Medivation, Inc. In 2009 this drug was in phase III clinical trials for use in Alzheimer's disease, and also phase II clinical trials for use in Huntington's disease. In March 2010, the results of a clinical trial phase III were released; the investigational Alzheimer's disease drug Dimebon failed in the pivotal CONNECTION trial of patients with mild-to-moderate disease. With CONCERT, the remaining Pfizer and Medivation Phase III trial for Dimebon (latrepirdine) in Alzheimer's disease failed in 2012, effectively ending the development in this indication.

In another experiment using a rat model of Alzheimer's disease, it was demonstrated that systemic administration of hypothalamic proline-rich peptide (PRP)-1 offers neuroprotective effects and can prevent neurodegeneration in hippocampus amyloid-beta 25–35. This suggests that there could be therapeutic value to PRP-1.

Other avenues of investigation

Protein degradation offers therapeutic options both in preventing the synthesis and degradation of irregular proteins. There is also interest in upregulating autophagy to help clear protein aggregates implicated in neurodegeneration. Both of these options involve very complex pathways that we are only beginning to understand.

The goal of immunotherapy is to enhance aspects of the immune system. Both active and passive vaccinations have been proposed for Alzheimer's disease and other conditions; however, more research must be done to prove safety and efficacy in humans.

A current therapeutic target for the treatment of Alzheimer's disease is the protease β-secretase, which is involved in the amyloidogenic processing pathway that leads to the pathological accumulation of proteins in the brain. When the gene that encodes for amyloid precursor protein (APP) is spliced by α-secretase rather than β-secretase, the toxic protein β amyloid is not produced. Targeted inhibition of β-secretase can potentially prevent the neuronal death that is responsible for the symptoms of Alzheimer's disease.