Mathematical model

From Wikipedia, the free encyclopedia

A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, biology, earth science, chemistry) and engineering disciplines (such as computer science, electrical engineering), as well as in the social sciences (such as economics, psychology, sociology, political science).

A model may help to explain a system and to study the effects of different components, and to make predictions about behaviour.

Elements of a mathematical model

Mathematical models can take many forms, including dynamical systems, statistical models, differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures. In general, mathematical models may include logical models. In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements often leads to important advances as better theories are developed. 

In the physical sciences, a traditional mathematical model contains most of the following elements:

1. Governing equations
2. Supplementary sub-models
   1. Defining equations
   2. Constitutive equations
3. Assumptions and constraints
   1. Initial and boundary conditions
   2. Classical constraints and kinematic equations

Classifications

Mathematical models are usually composed of relationships and variables. Relationships can be described by operators, such as algebraic operators, functions, differential operators, etc. Variables are abstractions of system parameters of interest, that can be quantified. Several classification criteria can be used for mathematical models according to their structure:

• Linear vs. nonlinear: If all the operators in a mathematical model exhibit linearity, the resulting mathematical model is defined as linear. A model is considered to be nonlinear otherwise. The definition of linearity and nonlinearity is dependent on context, and linear models may have nonlinear expressions in them. For example, in a statistical linear model, it is assumed that a relationship is linear in the parameters, but it may be nonlinear in the predictor variables. Similarly, a differential equation is said to be linear if it can be written with linear differential operators, but it can still have nonlinear expressions in it. In a mathematical programming model, if the objective functions and constraints are represented entirely by linear equations, then the model is regarded as a linear model. If one or more of the objective functions or constraints are represented with a nonlinear equation, then the model is known as a nonlinear model.
  Nonlinearity, even in fairly simple systems, is often associated with phenomena such as chaos and irreversibility. Although there are exceptions, nonlinear systems and models tend to be more difficult to study than linear ones. A common approach to nonlinear problems is linearization, but this can be problematic if one is trying to study aspects such as irreversibility, which are strongly tied to nonlinearity.
• Static vs. dynamic: A dynamic model accounts for time-dependent changes in the state of the system, while a static (or steady-state) model calculates the system in equilibrium, and thus is time-invariant. Dynamic models typically are represented by differential equations or difference equations.
• Explicit vs. implicit: If all of the input parameters of the overall model are known, and the output parameters can be calculated by a finite series of computations, the model is said to be explicit. But sometimes it is the output parameters which are known, and the corresponding inputs must be solved for by an iterative procedure, such as Newton's method (if the model is linear) or Broyden's method (if non-linear). In such a case the model is said to be implicit. For example, a jet engine's physical properties such as turbine and nozzle throat areas can be explicitly calculated given a design thermodynamic cycle (air and fuel flow rates, pressures, and temperatures) at a specific flight condition and power setting, but the engine's operating cycles at other flight conditions and power settings cannot be explicitly calculated from the constant physical properties.
• Discrete vs. continuous: A discrete model treats objects as discrete, such as the particles in a molecular model or the states in a statistical model; while a continuous model represents the objects in a continuous manner, such as the velocity field of fluid in pipe flows, temperatures and stresses in a solid, and electric field that applies continuously over the entire model due to a point charge.
• Deterministic vs. probabilistic (stochastic): A deterministic model is one in which every set of variable states is uniquely determined by parameters in the model and by sets of previous states of these variables; therefore, a deterministic model always performs the same way for a given set of initial conditions. Conversely, in a stochastic model—usually called a "statistical model"—randomness is present, and variable states are not described by unique values, but rather by probability distributions.
• Deductive, inductive, or floating: A deductive model is a logical structure based on a theory. An inductive model arises from empirical findings and generalization from them. The floating model rests on neither theory nor observation, but is merely the invocation of expected structure. Application of mathematics in social sciences outside of economics has been criticized for unfounded models. Application of catastrophe theory in science has been characterized as a floating model.

Significance in the natural sciences

Mathematical models are of great importance in the natural sciences, particularly in physics. Physical theories are almost invariably expressed using mathematical models.

Throughout history, more and more accurate mathematical models have been developed. Newton's laws accurately describe many everyday phenomena, but at certain limits theory of relativity and quantum mechanics must be used. Though even these theories can't model or explain all phenomena themselves or together, such as black holes. It is possible to obtain the less accurate models in appropriate limits, for example relativistic mechanics reduces to Newtonian mechanics at speeds much less than the speed of light. Quantum mechanics reduces to classical physics when the quantum numbers are high. For example, the de Broglie wavelength of a tennis ball is insignificantly small, so classical physics is a good approximation to use in this case. 

It is common to use idealized models in physics to simplify things. Massless ropes, point particles, ideal gases and the particle in a box are among the many simplified models used in physics. The laws of physics are represented with simple equations such as Newton's laws, Maxwell's equations and the Schrödinger equation. These laws are a basis for making mathematical models of real situations. Many real situations are very complex and thus modeled approximate on a computer, a model that is computationally feasible to compute is made from the basic laws or from approximate models made from the basic laws. For example, molecules can be modeled by molecular orbital models that are approximate solutions to the Schrödinger equation. In engineering, physics models are often made by mathematical methods such as finite element analysis.

Different mathematical models use different geometries that are not necessarily accurate descriptions of the geometry of the universe. Euclidean geometry is much used in classical physics, while special relativity and general relativity are examples of theories that use geometries which are not Euclidean.

Some applications

Since prehistorical times simple models such as maps and diagrams have been used.

Often when engineers analyze a system to be controlled or optimized, they use a mathematical model. In analysis, engineers can build a descriptive model of the system as a hypothesis of how the system could work, or try to estimate how an unforeseeable event could affect the system. Similarly, in control of a system, engineers can try out different control approaches in simulations.

A mathematical model usually describes a system by a set of variables and a set of equations that establish relationships between the variables. Variables may be of many types; real or integer numbers, boolean values or strings, for example. The variables represent some properties of the system, for example, measured system outputs often in the form of signals, timing data, counters, and event occurrence (yes/no). The actual model is the set of functions that describe the relations between the different variables.

Building blocks

In business and engineering, mathematical models may be used to maximize a certain output. The system under consideration will require certain inputs. The system relating inputs to outputs depends on other variables too: decision variables, state variables, exogenous variables, and random variables.

Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known as parameters or constants. The variables are not independent of each other as the state variables are dependent on the decision, input, random, and exogenous variables. Furthermore, the output variables are dependent on the state of the system (represented by the state variables).

Objectives and constraints of the system and its users can be represented as functions of the output variables or state variables. The objective functions will depend on the perspective of the model's user. Depending on the context, an objective function is also known as an index of performance, as it is some measure of interest to the user. Although there is no limit to the number of objective functions and constraints a model can have, using or optimizing the model becomes more involved (computationally) as the number increases. 

For example, economists often apply linear algebra when using input-output models. Complicated mathematical models that have many variables may be consolidated by use of vectors where one symbol represents several variables.

A priori information

To analyse something with a typical "black box approach", only the behavior of the stimulus/response will be accounted for, to infer the (unknown) box. The usual representation of this black box system is a data flow diagram centered in the box.

 

Mathematical modeling problems are often classified into black box or white box models, according to how much a priori information on the system is available. A black-box model is a system of which there is no a priori information available. A white-box model (also called glass box or clear box) is a system where all necessary information is available. Practically all systems are somewhere between the black-box and white-box models, so this concept is useful only as an intuitive guide for deciding which approach to take.

Usually it is preferable to use as much a priori information as possible to make the model more accurate. Therefore, the white-box models are usually considered easier, because if you have used the information correctly, then the model will behave correctly. Often the a priori information comes in forms of knowing the type of functions relating different variables. For example, if we make a model of how a medicine works in a human system, we know that usually the amount of medicine in the blood is an exponentially decaying function. But we are still left with several unknown parameters; how rapidly does the medicine amount decay, and what is the initial amount of medicine in blood? This example is therefore not a completely white-box model. These parameters have to be estimated through some means before one can use the model.

In black-box models one tries to estimate both the functional form of relations between variables and the numerical parameters in those functions. Using a priori information we could end up, for example, with a set of functions that probably could describe the system adequately. If there is no a priori information we would try to use functions as general as possible to cover all different models. An often used approach for black-box models are neural networks which usually do not make assumptions about incoming data. Alternatively the NARMAX (Nonlinear AutoRegressive Moving Average model with eXogenous inputs) algorithms which were developed as part of nonlinear system identification can be used to select the model terms, determine the model structure, and estimate the unknown parameters in the presence of correlated and nonlinear noise. The advantage of NARMAX models compared to neural networks is that NARMAX produces models that can be written down and related to the underlying process, whereas neural networks produce an approximation that is opaque.

Subjective information

Sometimes it is useful to incorporate subjective information into a mathematical model. This can be done based on intuition, experience, or expert opinion, or based on convenience of mathematical form. Bayesian statistics provides a theoretical framework for incorporating such subjectivity into a rigorous analysis: we specify a prior probability distribution (which can be subjective), and then update this distribution based on empirical data.

An example of when such approach would be necessary is a situation in which an experimenter bends a coin slightly and tosses it once, recording whether it comes up heads, and is then given the task of predicting the probability that the next flip comes up heads. After bending the coin, the true probability that the coin will come up heads is unknown; so the experimenter would need to make a decision (perhaps by looking at the shape of the coin) about what prior distribution to use. Incorporation of such subjective information might be important to get an accurate estimate of the probability.

Complexity

In general, model complexity involves a trade-off between simplicity and accuracy of the model. Occam's razor is a principle particularly relevant to modeling, its essential idea being that among models with roughly equal predictive power, the simplest one is the most desirable. While added complexity usually improves the realism of a model, it can make the model difficult to understand and analyze, and can also pose computational problems, including numerical instability. Thomas Kuhn argues that as science progresses, explanations tend to become more complex before a paradigm shift offers radical simplification.

For example, when modeling the flight of an aircraft, we could embed each mechanical part of the aircraft into our model and would thus acquire an almost white-box model of the system. However, the computational cost of adding such a huge amount of detail would effectively inhibit the usage of such a model. Additionally, the uncertainty would increase due to an overly complex system, because each separate part induces some amount of variance into the model. It is therefore usually appropriate to make some approximations to reduce the model to a sensible size. Engineers often can accept some approximations in order to get a more robust and simple model. For example, Newton's classical mechanics is an approximated model of the real world. Still, Newton's model is quite sufficient for most ordinary-life situations, that is, as long as particle speeds are well below the speed of light, and we study macro-particles only.

Training and tuning

Any model which is not pure white-box contains some parameters that can be used to fit the model to the system it is intended to describe. If the modeling is done by an artificial neural network or other machine learning, the optimization of parameters is called training, while the optimization of model hyperparameters is called tuning and often uses cross-validation. In more conventional modeling through explicitly given mathematical functions, parameters are often determined by curve fitting.

Model evaluation

A crucial part of the modeling process is the evaluation of whether or not a given mathematical model describes a system accurately. This question can be difficult to answer as it involves several different types of evaluation.

Fit to empirical data

Usually the easiest part of model evaluation is checking whether a model fits experimental measurements or other empirical data. In models with parameters, a common approach to test this fit is to split the data into two disjoint subsets: training data and verification data. The training data are used to estimate the model parameters. An accurate model will closely match the verification data even though these data were not used to set the model's parameters. This practice is referred to as cross-validation in statistics.

Defining a metric to measure distances between observed and predicted data is a useful tool of assessing model fit. In statistics, decision theory, and some economic models, a loss function plays a similar role.

While it is rather straightforward to test the appropriateness of parameters, it can be more difficult to test the validity of the general mathematical form of a model. In general, more mathematical tools have been developed to test the fit of statistical models than models involving differential equations. Tools from non-parametric statistics can sometimes be used to evaluate how well the data fit a known distribution or to come up with a general model that makes only minimal assumptions about the model's mathematical form.

Scope of the model

Assessing the scope of a model, that is, determining what situations the model is applicable to, can be less straightforward. If the model was constructed based on a set of data, one must determine for which systems or situations the known data is a "typical" set of data.

The question of whether the model describes well the properties of the system between data points is called interpolation, and the same question for events or data points outside the observed data is called extrapolation.

As an example of the typical limitations of the scope of a model, in evaluating Newtonian classical mechanics, we can note that Newton made his measurements without advanced equipment, so he could not measure properties of particles travelling at speeds close to the speed of light. Likewise, he did not measure the movements of molecules and other small particles, but macro particles only. It is then not surprising that his model does not extrapolate well into these domains, even though his model is quite sufficient for ordinary life physics.

Philosophical considerations

Many types of modeling implicitly involve claims about causality. This is usually (but not always) true of models involving differential equations. As the purpose of modeling is to increase our understanding of the world, the validity of a model rests not only on its fit to empirical observations, but also on its ability to extrapolate to situations or data beyond those originally described in the model. One can think of this as the differentiation between qualitative and quantitative predictions. One can also argue that a model is worthless unless it provides some insight which goes beyond what is already known from direct investigation of the phenomenon being studied.

An example of such criticism is the argument that the mathematical models of optimal foraging theory do not offer insight that goes beyond the common-sense conclusions of evolution and other basic principles of ecology.

Examples

• One of the popular examples in computer science is the mathematical models of various machines, an example is the deterministic finite automaton (DFA) which is defined as an abstract mathematical concept, but due to the deterministic nature of a DFA, it is implementable in hardware and software for solving various specific problems. For example, the following is a DFA M with a binary alphabet, which requires that the input contains an even number of 0s.

The state diagram for M

M = (Q, Σ, δ, q₀, F) where

• Q = {S₁, S₂},
• Σ = {0, 1},
• q₀ = S₁,
• F = {S₁}, and
• δ is defined by the following state transition table:

	0	1
S1	S2	S1
S2	S1	S2

The state S₁ represents that there has been an even number of 0s in the input so far, while S₂ signifies an odd number. A 1 in the input does not change the state of the automaton. When the input ends, the state will show whether the input contained an even number of 0s or not. If the input did contain an even number of 0s, M will finish in state S₁, an accepting state, so the input string will be accepted.

The language recognized by M is the regular language given by the regular expression 1*( 0 (1*) 0 (1*) )*, where "*" is the Kleene star, e.g., 1* denotes any non-negative number (possibly zero) of symbols "1".

• Many everyday activities carried out without a thought are uses of mathematical models. A geographical map projection of a region of the earth onto a small, plane surface is a model^[7] which can be used for many purposes such as planning travel.
• Another simple activity is predicting the position of a vehicle from its initial position, direction and speed of travel, using the equation that distance traveled is the product of time and speed. This is known as dead reckoning when used more formally. Mathematical modeling in this way does not necessarily require formal mathematics; animals have been shown to use dead reckoning.
• Population Growth. A simple (though approximate) model of population growth is the Malthusian growth model. A slightly more realistic and largely used population growth model is the logistic function, and its extensions.
• Model of a particle in a potential-field. In this model we consider a particle as being a point of mass which describes a trajectory in space which is modeled by a function giving its coordinates in space as a function of time. The potential field is given by a function ${\displaystyle V\!:\mathbb {R} ^{3}\!\rightarrow \mathbb {R} }$ and the trajectory, that is a function ${\displaystyle \mathbf {r} \!:\mathbb {R} \rightarrow \mathbb {R} ^{3}}$, is the solution of the differential equation:

${\displaystyle -{\frac {\mathrm {d} ^{2}\mathbf {r} (t)}{\mathrm {d} t^{2}}}m={\frac {\partial V[\mathbf {r} (t)]}{\partial x}}\mathbf {\hat {x}} +{\frac {\partial V[\mathbf {r} (t)]}{\partial y}}\mathbf {\hat {y}} +{\frac {\partial V[\mathbf {r} (t)]}{\partial z}}\mathbf {\hat {z}} ,}$

that can be written also as:

${\displaystyle m{\frac {\mathrm {d} ^{2}\mathbf {r} (t)}{\mathrm {d} t^{2}}}=-\nabla V[\mathbf {r} (t)].}$

Note this model assumes the particle is a point mass, which is certainly known to be false in many cases in which we use this model; for example, as a model of planetary motion.

• Model of rational behavior for a consumer. In this model we assume a consumer faces a choice of n commodities labeled 1,2,...,n each with a market price p₁, p₂,..., p_n. The consumer is assumed to have an ordinal utility function U (ordinal in the sense that only the sign of the differences between two utilities, and not the level of each utility, is meaningful), depending on the amounts of commodities x₁, x₂,..., x_n consumed. The model further assumes that the consumer has a budget M which is used to purchase a vector x₁, x₂,..., x_n in such a way as to maximize U(x₁, x₂,..., x_n). The problem of rational behavior in this model then becomes an optimization problem, that is:

${\displaystyle \max U(x_{1},x_{2},\ldots ,x_{n})}$

subject to:

${\displaystyle \sum _{i=1}^{n}p_{i}x_{i}\leq M.}$

${\displaystyle x_{i}\geq 0\;\;\;\forall i\in \{1,2,\ldots ,n\}}$

This model has been used in a wide variety of economic contexts, such as in general equilibrium theory to show existence and Pareto efficiency of economic equilibria.

• Neighbour-sensing model explains the mushroom formation from the initially chaotic fungal network.
• Computer science: models in Computer Networks, data models, surface model,...
• Mechanics: movement of rocket model,...

Cerebral hemisphere

From Wikipedia, the free encyclopedia

Cerebral hemisphere
Human brain seen from front.
Right cerebral hemisphere   Left cerebral hemisphere
Details
Identifiers
Latin	Hemisphaerium cerebri
NeuroNames	241
NeuroLex ID	birnlex_1796
TA	A14.1.09.002
FMA	61817

The vertebrate cerebrum (brain) is formed by two cerebral hemispheres that are separated by a groove, the longitudinal fissure. The brain can thus be described as being divided into left and right cerebral hemispheres. Each of these hemispheres has an outer layer of grey matter, the cerebral cortex, that is supported by an inner layer of white matter. In eutherian (placental) mammals, the hemispheres are linked by the corpus callosum, a very large bundle of nerve fibers. Smaller commissures, including the anterior commissure, the posterior commissure and the fornix, also join the hemispheres and these are also present in other vertebrates. These commissures transfer information between the two hemispheres to coordinate localized functions.

There are three known poles of the cerebral hemispheres: the occipital pole, the frontal pole, and the temporal pole.

The central sulcus is a prominent fissure which separates the parietal lobe from the frontal lobe and the primary motor cortex from the primary somatosensory cortex.

Macroscopically the hemispheres are roughly mirror images of each other, with only subtle differences, such as the Yakovlevian torque seen in the human brain, which is a slight warping of the right side, bringing it just forward of the left side. On a microscopic level, the cytoarchitecture of the cerebral cortex, shows the functions of cells, quantities of neurotransmitter levels and receptor subtypes to be markedly asymmetrical between the hemispheres. However, while some of these hemispheric distribution differences are consistent across human beings, or even across some species, many observable distribution differences vary from individual to individual within a given species.

Structure

Each cerebral hemisphere has an outer layer of cerebral cortex which is of grey matter and in the interior of the cerebral hemispheres is an inner layer or core of white matter known as the centrum semiovale. The interior portion of the hemispheres of the cerebrum includes the lateral ventricles, the basal ganglia, and the white matter.

Poles

Poles of cerebral hemispheres

 

There are three poles of the cerebrum, the occipital pole, the frontal pole, and the temporal pole. The occipital pole is the posterior end of each occipital lobe in each hemisphere. It is more pointed than the rounder frontal pole. The frontal pole is at the frontmost part of the frontal lobe in each hemisphere, and is more rounded than the occipital pole. The temporal pole is located between the frontal and occipital poles, and sits in the anterior part of middle cranial fossa in each temporal lobe.

Composition

If the upper part of either hemisphere is removed, at a level about 1.25 cm above the corpus callosum, the central white matter will be exposed as an oval-shaped area, the centrum semiovale, surrounded by a narrow convoluted margin of gray substance, and studded with numerous minute red dots (puncta vasculosa), produced by the escape of blood from divided blood vessels.

If the remaining portions of the hemispheres be slightly drawn apart a broad band of white substance, the corpus callosum, will be observed, connecting them at the bottom of the longitudinal fissure; the margins of the hemispheres which overlap the corpus callosum are called the labia cerebri.

Each labium is part of the cingulate gyrus already described; and the groove between it and the upper surface of the corpus callosum is termed the callosal sulcus.

If the hemispheres are sliced off to a level with the upper surface of the corpus callosum, the white substance of that structure will be seen connecting the two hemispheres.

The large expanse of medullary matter now exposed, surrounded by the convoluted margin of gray substance, is called the centrum semiovale. The blood supply to the centrum semiovale is from the superficial middle cerebral artery. The cortical branches of this artery descend to provide blood to the centrum semiovale.

Development

The cerebral hemispheres are derived from the telencephalon. They arise five weeks after conception as bilateral invaginations of the walls. The hemispheres grow round in a C-shape and then back again, pulling all structures internal to the hemispheres (such as the ventricles) with them. The intraventricular foramina (also called the foramina of Monro) allows communication with the lateral ventricles. The choroid plexus is formed from ependymal cells and vascular mesenchyme.

Function

Hemisphere lateralization

Broad generalizations are often made in popular psychology about certain functions (e.g. logic, creativity) being lateralized, that is, located in the right or left side of the brain. These claims are often inaccurate, as most brain functions are actually distributed across both hemispheres. Most scientific evidence for asymmetry relates to low-level perceptual functions rather than the higher-level functions popularly discussed (e.g. subconscious processing of grammar, not "logical thinking" in general). In addition to this lateralization of some functions, the low-level representations also tend to represent the contralateral side of the body.

The best example of an established lateralization is that of Broca's and Wernicke's Areas (language) where both are often found exclusively on the left hemisphere. These areas frequently correspond to handedness however, meaning the localization of these areas is regularly found on the hemisphere opposite to the dominant hand. Function lateralization such as semantics, prosodic, intonation, accentuation, prosody, etc. has since been called into question and largely been found to have a neuronal basis in both hemispheres.

Cerebral hemispheres of a human embryo at 8 weeks.

 

Perceptual information is processed in both hemispheres, but is laterally partitioned: information from each side of the body is sent to the opposite hemisphere (visual information is partitioned somewhat differently, but still lateralized). Similarly, motor control signals sent out to the body also come from the hemisphere on the opposite side. Thus, hand preference (which hand someone prefers to use) is also related to hemisphere lateralization.

In some aspects, the hemispheres are asymmetrical; the right side is slightly bigger. There are higher levels of the neurotransmitter norepinephrine on the right and higher levels of dopamine on the left. There is more white matter (longer axons) on the right and more grey matter (cell bodies) on the left.

Linear reasoning functions of language such as grammar and word production are often lateralized to the left hemisphere of the brain. In contrast, holistic reasoning functions of language such as intonation and emphasis are often lateralized to the right hemisphere of the brain. Other integrative functions such as intuitive or heuristic arithmetic, binaural sound localization, etc. seem to be more bilaterally controlled.

Clinical significance

Infarcts of the centrum ovale can occur.

As a treatment for epilepsy the corpus callosum may be severed to cut the major connection between the hemispheres in a procedure known as a corpus callosotomy. 

A hemispherectomy is the removal or disabling of one of the hemispheres of the brain. This is a rare procedure used in some extreme cases of seizures which are unresponsive to other treatments.

Broca's area

From Wikipedia, the free encyclopedia

 

Broca's area
Broca's area is made up of Brodmann areas 44 (pars opercularis) and 45 (pars triangularis)
Broca's area (shown in red)
Details
Part of	Frontal lobe
Artery	Middle cerebral
Vein	Superior sagittal sinus
Identifiers
MeSH	D065711
NeuroNames	2062
FMA	242176

Broca's area, or the Broca area , is a region in the frontal lobe of the dominant hemisphere, usually the left, of the brain with functions linked to speech production.

Language processing has been linked to Broca's area since Pierre Paul Broca reported impairments in two patients. They had lost the ability to speak after injury to the posterior inferior frontal gyrus (pars triangularis) (BA45) of the brain. Since then, the approximate region he identified has become known as Broca's area, and the deficit in language production as Broca's aphasia, also called expressive aphasia. Broca's area is now typically defined in terms of the pars opercularis and pars triangularis of the inferior frontal gyrus, represented in Brodmann's cytoarchitectonic map as Brodmann area 44 and Brodmann area 45 of the dominant hemisphere.

Functional magnetic resonance imaging has shown language processing to also involve the third part of the inferior frontal gyrus the pars orbitalis, as well as the ventral part of BA6 and these are now often included in a larger area called Broca's region.

Studies of chronic aphasia have implicated an essential role of Broca's area in various speech and language functions. Further, fMRI studies have also identified activation patterns in Broca's area associated with various language tasks. However, slow destruction of the Broca's area by brain tumors can leave speech relatively intact, suggesting its functions can shift to nearby areas in the brain.

Structure

Broca's area is often identified by visual inspection of the topography of the brain either by macrostructural landmarks such as sulci or by the specification of coordinates in a particular reference space. The currently used Talairach and Tournoux atlas projects Brodmann's cytoarchitectonic map onto a template brain. Because Brodmann's parcelation was based on subjective visual inspection of cytoarchitectonic borders and also Brodmann analyzed only one hemisphere of one brain, the result is imprecise. Further, because of considerable variability across brains in terms of shape, size, and position relative to sulcal and gyral structure, a resulting localization precision is limited.

Nevertheless, Broca's area in the left hemisphere and its homologue in the right hemisphere are designations usually used to refer to the triangular part of inferior frontal gyrus (PTr) and the opercular part of inferior frontal gyrus (POp). The PTr and POp are defined by structural landmarks that only probabilistically divide the inferior frontal gyrus into anterior and posterior cytoarchitectonic areas of 45 and 44, respectively, by Brodmann's classification scheme.

Area 45 receives more afferent connections from the prefrontal cortex, the superior temporal gyrus, and the superior temporal sulcus, compared to area 44, which tends to receive more afferent connections from motor, somatosensory, and inferior parietal regions.

The differences between area 45 and 44 in cytoarchitecture and in connectivity suggest that these areas might perform different functions. Indeed, recent neuroimaging studies have shown that the PTr and Pop, corresponding to areas 45 and 44, respectively, play different functional roles in the human with respect to language comprehension and action recognition/understanding.

Functions

Language comprehension

For a long time, it was assumed that the role of Broca's area was more devoted to language production than language comprehension. However, there is evidence to demonstrate that Broca's area also plays a significant role in language comprehension. Patients with lesions in Broca's area who exhibit agrammatical speech production also show inability to use syntactic information to determine the meaning of sentences. Also, a number of neuroimaging studies have implicated an involvement of Broca's area, particularly of the pars opercularis of the left inferior frontal gyrus, during the processing of complex sentences. Further, it has recently been found in functional magnetic resonance imaging (fMRI) experiments involving highly ambiguous sentences result in a more activated inferior frontal gyrus. Therefore, the activity level in the inferior frontal gyrus and the level of lexical ambiguity are directly proportional to each other, because of the increased retrieval demands associated with highly ambiguous content.

There is also specialisation for particular aspects of comprehension within Broca's area. Work by Devlin et al. (2003) showed in a repetitive transcranial magnetic stimulation (rTMS) study that there was an increase in reaction times when performing a semantic task under rTMS aimed at the pars triangularis (situated in the anterior part of Broca's area). The increase in reaction times is indicative that that particular area is responsible for processing that cognitive function. Disrupting these areas via TMS disrupts computations performed in the areas leading to an increase in time needed to perform the computations (reflected in reaction times). Later work by Nixon et al. (2004) showed that when the pars opercularis (situated in the posterior part of Broca's area) was stimulated under rTMS there was an increase in reaction times in a phonological task. Gough et al. (2005) performed an experiment combining elements of these previous works in which both phonological and semantic tasks were performed with rTMS stimulation directed at either the anterior or the posterior part of Broca's area. The results from this experiment conclusively distinguished anatomical specialisation within Broca's area for different components of language comprehension. Here the results showed that under rTMS stimulation:

• Semantic tasks only showed a decrease in reaction times when stimulation was aimed at the anterior part of Broca's area (where a decrease of 10% (50ms) was seen compared to a no-TMS control group)
• Phonological tasks showed a decrease in reaction times when stimulation was aimed at the posterior part of Broca's area (where a decrease of 6% (30ms) was seen compared to control)

To summarise, the work above shows anatomical specialisation in Broca's area for language comprehension, with the anterior part of Broca's area responsible for understanding the meaning of words (semantics) and the posterior part of Broca's area responsible for understanding how words sound (phonology).

Action recognition and production

Recent experiments have indicated that Broca's area is involved in various cognitive and perceptual tasks. One important contribution of Brodmann's area 44 is also found in the motor-related processes. Observation of meaningful hand shadows resembling moving animals activates frontal language area, demonstrating that Broca's area indeed plays a role in interpreting action of others. An activation of BA 44 was also reported during execution of grasping and manipulation.

Speech-associated gestures

It has been speculated that because speech-associated gestures could possibly reduce lexical or sentential ambiguity, comprehension should improve in the presence of speech-associated gestures. As a result of improved comprehension, the involvement of Broca's area should be reduced.

Many neuroimaging studies have also shown activation of Broca's area when representing meaningful arm gestures. A recent study has shown evidence that word and gesture are related at the level of translation of particular gesture aspects such as its motor goal and intention. This finding helps explain why, when this area is defective, those who use sign language also suffer from language deficits. This finding that aspects of gestures are translated in words within Broca's area also explains language development in terms of evolution. Indeed, many authors have proposed that speech evolved from a primitive communication that arose from gestures.

Speaking without Broca's area

Damage to Broca's area is commonly associated with telegraphic speech made up of content vocabulary. For example, a person with Broca's aphasia may say something like, "Drive, store. Mom." meaning to say, "My mom drove me to the store today." Therefore, the content of the information is correct, but the grammar and fluidity of the sentence is missing.

The essential role of the Broca's area in speech production has been questioned since it can be destroyed while leaving language nearly intact. In one case of a computer engineer, a slow-growing glioma tumor was removed. The tumor and the surgery destroyed the left inferior and middle frontal gyrus, the head of the caudate nucleus, the anterior limb of the internal capsule, and the anterior insula. However, there were minimal language problems three months after removal and the individual returned to his professional work. These minor problems include the inability to create syntactically complex sentences including more than two subjects, multiple causal conjunctions, or reported speech. These were explained by researchers as due to working memory problems. They also attributed his lack of problems to extensive compensatory mechanisms enabled by neural plasticity in the nearby cerebral cortex and a shift of some functions to the homologous area in the right hemisphere.

Clinical significance

Stuttering

A speech disorder known as stuttering is seen to be associated with underactivity in Broca's area.

Aphasia

Aphasia is an acquired language disorder affecting all modalities such as writing, reading, speaking, and listening and results from brain damage. It is often a chronic condition that creates changes in all areas of one's life.

Expressive aphasia vs. other aphasias

Patients with expressive aphasia, also known as Broca's aphasia, are individuals who know "what they want to say, they just cannot get it out". They are typically able to comprehend words, and sentences with a simple syntactic structure (see above), but are more or less unable to generate fluent speech. Other symptoms that may be present include problems with fluency, articulation, word-finding, word repetition, and producing and comprehending complex grammatical sentences, both orally and in writing.

This specific group of symptoms distinguishes those who have expressive aphasia from individuals with other types of aphasia. There are several distinct "types" of aphasia, and each type is characterized by a different set of language deficits. Although those who have expressive aphasia tend to retain good spoken language comprehension, other types of aphasia can render patients completely unable to understand any language at all, unable to understand any spoken language (auditory verbal agnosia), whereas still other types preserve language comprehension, but with deficits. People with expressive aphasia may struggle less with reading and writing than those with other types of aphasia. Although individuals with expressive aphasia tend to have a good ability to self-monitor their language output (they "hear what they say" and make corrections), other types of aphasics can seem entirely unaware of their language deficits. 

In the classical sense, expressive aphasia is the result of injury to Broca's area; it is often the case that lesions in specific brain areas cause specific, dissociable symptoms, although case studies show there is not always a one-to-one mapping between lesion location and aphasic symptoms. The correlation between damage to certain specific brain areas (usually in the left hemisphere) and the development of specific types of aphasia makes it possible to deduce (albeit very roughly) the location of a suspected brain lesion based only on the presence (and severity) of a certain type of aphasia, though this is complicated by the possibility that a patient may have damage to a number of brain areas and may exhibit symptoms of more than one type of aphasia. The examination of lesion data in order to deduce which brain areas are essential in the normal functioning of certain aspects of cognition is called the deficit-lesion method; this method is especially important in the branch of neuroscience known as aphasiology. Cognitive science - to be specific, cognitive neuropsychology - are branches of neuroscience that also make extensive use of the deficit-lesion method.

Major characteristics of different types of acute aphasia

Type of aphasia	Speech repetition	Naming	Auditory comprehension	Fluency
Expressive aphasia	Moderate–severe	Moderate–severe	Mild difficulty	Non-fluent, effortful, slow
Receptive aphasia	Mild–severe	Mild–severe	Defective	Fluent paraphasic
Conduction aphasia	Poor	Poor	Relatively good	Fluent
Mixed transcortical aphasia	Moderate	Poor	Poor	Non-fluent
Transcortical motor aphasia	Good	Mild–severe	Mild	Non-fluent
Transcortical sensory aphasia	Good	Moderate–severe	Poor	Fluent
Global aphasia	Poor	Poor	Poor	Non-fluent
Anomic aphasia	Mild	Moderate–severe	Mild	Fluent

Newer implications related to lesions in Broca's area

Since studies carried out in the late 1970's it has been understood that the relationship between Broca's area and Broca's aphasia is not as consistent as once thought. Lesions to Broca's area alone don't result in a Broca's aphasia, nor do Broca's aphasic patients necessarily have lesions in Broca's area. Lesions to Broca's area alone are known to produce just a transient mutism that resolves inside 3–6 weeks. This discovery suggests that Broca's area may be included in some aspect of verbalization or articulation; however, it does not address its part in sentence comprehension. Still, Broca's area frequently emerges in functional imaging studies of sentence processing. However, it also becomes activated in word-level tasks. This suggests that Broca’s area is not dedicated to sentence processing but supports a function common to both. In fact, Broca's area can show activation in such non-linguistic tasks as imagery of motion.

Considering the hypothesis that Broca's area may be most involved in articulation, its activation in all of these tasks may be due to subjects' covert articulation while formulating a response. Despite this caveat, a consensus seems to be forming that whatever role Broca's area may play, it may relate to known working memory functions of the frontal areas. (There is a wide distribution of Talairach coordinates reported in the functional imaging literature that are referred to as part of Broca's area.) The processing of a passive voice sentence, for example, may require working memory to assist in the temporary retention of information while other relevant parts of the sentence are being manipulated (i.e. to resolve the assignment of thematic roles to arguments). Miyake, Carpenter, and Just have proposed that sentence processing relies on such general verbal working memory mechanisms while Caplan and Waters consider Broca’s area to be involved in working memory specifically for syntactic processing. Friederici (2002) breaks Broca's area into its component regions and suggests that Brodmann's area 44 is involved in working memory for both phonological and syntactic structure. This area becomes active first for phonology and later for syntax as the time course for the comprehension process unfolds. Brodmann's area 45 together with Brodmann's area 47 is viewed as being specifically involved in working memory for semantic features and thematic structure where processes of syntactic reanalysis and repair are required. These areas come online after Brodmann's area 44 has finished its processing role and where comprehension of complex sentences must rely on general memory resources. All of these theories indicate a move towards a view that syntactic comprehension problems arise from a computational rather than a conceptual deficit. Newer theories are taking a more dynamic view of how the brain integrates different linguistic and cognitive components and are examining the time course of these operations.

Neurocognitive studies have already implicated frontal areas adjacent to Broca's area as important for working memory in non-linguistic as well as linguistic tasks. Cabeza and Nyberg's analysis of imaging studies of working memory supports the view that BA45/47 is recruited for selecting or comparing information, while BA9/46 might be more involved in the manipulation of information in working memory. Since large lesions are typically required to produce a Broca's aphasia, it is likely that these regions may also become compromised in some patients and may contribute to their comprehension deficits for complex morphosyntactic structures.

Broca's area: A key center in the linking phonemic sequences

Broca's area has been previously associated with a variety of processes, including phonological segmentation, syntactic processing, and unification, all of which involve segmenting and linking different types of linguistic information. Although repeating and reading single words do not engage semantic and syntactic processing, they do require an operation linking phonemic sequences with motor gestures. Findings indicate that this linkage is coordinated by Broca's area through reciprocal interactions with temporal and frontal cortices responsible for phonemic and articulatory representations, respectively, including interactions with motor cortex before the actual act of speech. Based on these unique findings, it has been proposed that Broca's area is not the seat of articulation per se, but rather is a key node in manipulating and forwarding neural information across large-scale cortical networks responsible for key components of speech production.

History

In a study published in 2007, the preserved brains of both Leborgne and Lelong (patients of Broca) were reinspected using high-resolution volumetric MRI. The purpose of this study was to scan the brains in three dimensions and to identify the extent of both cortical and subcortical lesions in more detail. The study also sought to locate the exact site of the lesion in the frontal lobe in relation to what is now called Broca's area with the extent of subcortical involvement.

Broca's patients

Louis Victor Leborgne (Tan)

Leborgne was a patient of Broca's. At 30 years old, he was almost completely unable to produce any words or phrases. He was able to repetitively produce only the word tan. After his death, a neurosyphilitic lesion was discovered on the surface of his left frontal lobe.

Lelong

Lelong was another patient of Broca's. He also exhibited reduced productive speech. He could only say five words, 'yes', 'no', 'three', 'always', and 'lelo' (a mispronunciation of his own name). A lesion within the lateral frontal lobe was discovered during Lelong's autopsy. Broca's previous patient, Leborgne, had this lesion in the same area of his frontal lobe. These two cases led Broca to believe that speech was localized to this particular area.

MRI findings

Examination of the brains of Broca's two historic patients with high-resolution MRI has produced several interesting findings. First, the MRI findings suggest that other areas besides Broca's area may also have contributed to the patients' reduced productive speech. This finding is significant because it has been found that, though lesions to Broca's area alone can possibly cause temporary speech disruption, they do not result in severe speech arrest. Therefore, there is a possibility that the aphasia denoted by Broca as an absence of productive speech also could have been influenced by the lesions in the other region. Another finding is that the region, which was once considered to be critical for speech by Broca, is not precisely the same region as what is now known as Broca's area. This study provides further evidence to support the claim that language and cognition are far more complicated than once thought and involve various networks of brain regions.

Evolution of language

The pursuit of a satisfying theory that addresses the origin of language in humans has led to the consideration of a number of evolutionary "models". These models attempt to show how modern language might have evolved, and a common feature of many of these theories is the idea that vocal communication was initially used to complement a far more dominant mode of communication through gesture. Human language might have evolved as the "evolutionary refinement of an implicit communication system already present in lower primates, based on a set of hand/mouth goal-directed action representations."

"Hand/mouth goal-directed action representations" is another way of saying "gestural communication", "gestural language", or "communication through body language". The recent finding that Broca's area is active when people are observing others engaged in meaningful action is evidence in support of this idea. It was hypothesized that a precursor to the modern Broca's area was involved in translating gestures into abstract ideas by interpreting the movements of others as meaningful action with an intelligent purpose. It is argued that over time the ability to predict the intended outcome and purpose of a set of movements eventually gave this area the capability to deal with truly abstract ideas, and therefore (eventually) became capable of associating sounds (words) with abstract meanings. The observation that frontal language areas are activated when people observe Hand Shadows is further evidence that human language may have evolved from existing neural substrates that evolved for the purpose of gesture recognition. The study, therefore, claims that Broca's area is the "motor center for speech", which assembles and decodes speech sounds in the same way it interprets body language and gestures. Consistent with this idea is that the neural substrate that regulated motor control in the common ancestor of apes and humans was most likely modified to enhance cognitive and linguistic ability. Studies of speakers of American Sign Language and English suggest that the human brain recruited systems that had evolved to perform more basic functions much earlier; these various brain circuits, according to the authors, were tapped to work together in creating language.

Another recent finding has showed significant areas of activation in subcortical and neocortical areas during the production of communicative manual gestures and vocal signals in chimpanzees. Further, the data indicating that chimpanzees intentionally produce manual gestures as well as vocal signals to communicate with humans suggests that the precursors to human language are present at both the behavioral and neuronanatomical levels. More recently, the neocortical distribution of activity-dependent gene expression in marmosets provided direct evidence that the ventrolateral prefrontal cortex, which comprises Broca's area in humans and has been associated with auditory processing of species-specific vocalizations and orofacial control in macaques, is engaged during vocal output in a New World monkey. These findings putatively set the origin of vocalization-related neocortical circuits to at least 35 million years ago, when the Old and New World monkey lineages split.

Wernicke's area

From Wikipedia, the free encyclopedia

 

Wernicke's area
Wernicke's area is located in the temporal lobe, shown here in grey
Details
Location	Temporal lobe of the dominant cerebral hemisphere.
Artery	Branches from the middle cerebral artery
Identifiers
MeSH	D065813
NeuroNames	1233
NeuroLex ID	nlx_144087
FMA	242178

Wernicke's area (/ˈvɛərnɪkə/ or /ˈvɛərnɪki/; German: [ˈvɛʁnɪkə]), also called Wernicke's speech area, is one of the two parts of the cerebral cortex that are linked to speech (the other is Broca's area). It is involved in the comprehension of written and spoken language (in contrast to Broca's area that is involved in the production of language). It is traditionally thought to be in Brodmann area 39,40, which is located in the superior temporal lobe in the dominant cerebral hemisphere (which is the left hemisphere in about 95% of right handed individuals and 60% of left handed individuals).

Damage caused to Wernicke's area results in receptive, fluent aphasia. This means that the person with aphasia will be able to fluently connect words, but the phrases will lack meaning. This is unlike non-fluent aphasia, in which the person will use meaningful words, but in a non-fluent, telegraphic manner.

Structure

Wernicke's area is classically located in the posterior section of the superior temporal gyrus (STG) in the (most commonly) left cerebral hemisphere. This area encircles the auditory cortex on the lateral sulcus (the part of the brain where the temporal lobe and parietal lobe meet). This area is neuroanatomically described as the posterior part of Brodmann area 22.

However, there is an absence of consistent definitions as to the location. Some identify it with the unimodal auditory association in the superior temporal gyrus anterior to the primary auditory cortex (the anterior part of BA 22). This is the site most consistently implicated in auditory word recognition by functional brain imaging experiments. Others include also adjacent parts of the heteromodal cortex in BA 39 and BA40 in the parietal lobe.

While previously thought to connect Wernicke's area and Broca's area, new research demonstrates that the arcuate fasciculus instead connects to posterior receptive areas with premotor/motor areas, and not to Broca's area. Consistent with the word recognition site identified in brain imaging, the uncinate fasciculus connects anterior superior temporal regions with Broca's area.

Function

Right homologous area

Research using Transcranial magnetic stimulation suggests that the area corresponding to the Wernicke’s area in the non-dominant cerebral hemisphere has a role in processing and resolution of subordinate meanings of ambiguous words—such as ‘‘river’’ when given the ambiguous word "bank." In contrast, the Wernicke's area in the dominant hemisphere processes dominant word meanings (‘‘teller’’ given ‘‘bank’’).

Modern views

Neuroimaging suggests the functions earlier attributed to Wernicke's area occur more broadly in the temporal lobe and indeed happen also in Broca's area.

“

There are some suggestions that middle and inferior temporal gyri and basal temporal cortex reflect lexical processing ... there is consensus that the STG from rostral to caudal fields and the STS constitute the neural tissue in which many of the critical computations for speech recognition are executed ... aspects of Broca’s area (Brodmann areas 44 and 45) are also regularly implicated in speech processing. ... the range of areas implicated in speech processing go well beyond the classical language areas typically mentioned for speech; the vast majority of textbooks still state that this aspect of perception and language processing occurs in Wernicke’s area (the posterior third of the STG).

”

Support for a broad range of speech processing areas was furthered by a recent study caried out at the University of Rochester in which American Sign Language native speakers were subject to MRI while interpreting sentences that identified a relationship using either syntax (relationship is determined by the word order) or inflection (relationship is determined by physical motion of "moving hands through space or signing on one side of the body"). Distinct areas of the brain were activated with the frontal cortex (associated with ability to put information into sequences) being more active in the syntax condition and the temporal lobes (associated with dividing information into its constituent parts) being more active in the inflection condition. However, these areas are not mutually exclusive and show a large amount of overlap. These findings imply that while speech processing is a very complex process, the brain may be using fairly basic, preexisting computational methods.

Clinical significance

Human brain with Wernicke's area highlighted in red

Aphasia

Wernicke's area is named after Carl Wernicke, a German neurologist and psychiatrist who, in 1874, hypothesized a link between the left posterior section of the superior temporal gyrus and the reflexive mimicking of words and their syllables that associated the sensory and motor images of spoken words. He did this on the basis of the location of brain injuries that caused aphasia. Receptive aphasia in which such abilities are preserved is also known as Wernicke's aphasia. In this condition there is a major impairment of language comprehension, while speech retains a natural-sounding rhythm and a relatively normal syntax. Language as a result is largely meaningless (a condition sometimes called fluent or jargon aphasia).

While neuroimaging and lesion evidence generally support the idea that malfunction of or damage to Wernicke's area is common in people with receptive aphasia, this is not always so. Some people may use the right hemisphere for language, and isolated damage of Wernicke's area cortex (sparing white matter and other areas) may not cause severe receptive aphasia. Even when patients with Wernicke's area lesions have comprehension deficits, these are usually not restricted to language processing alone. For example, one study found that patients with posterior lesions also had trouble understanding nonverbal sounds like animal and machine noises. In fact, for Wernicke's area, the impairments in nonverbal sounds were statistically stronger than for verbal sounds.