Logistic function

From Wikipedia, the free encyclopedia

Standard logistic sigmoid function i.e.

A logistic function or logistic curve is a common "S" shape (sigmoid curve), with equation:

${\displaystyle f(x)={\frac {L}{1+e^{-k(x-x_{0})}}}}$

where

• e = the natural logarithm base (also known as Euler's number),
• x₀ = the x-value of the sigmoid's midpoint,
• L = the curve's maximum value, and
• k = the logistic growth rate or steepness of the curve.

For values of x in the domain of real numbers from −∞ to +∞, the S-curve shown on the right is obtained, with the graph of f approaching L as x approaches +∞ and approaching zero as x approaches −∞. 

The logistic function finds applications in a range of fields, including artificial neural networks, biology (especially ecology), biomathematics, chemistry, demography, economics, geoscience, mathematical psychology, probability, sociology, political science, linguistics, and statistics.

History

Original image of a logistic curve, contrasted with a logarithmic curve

 

The logistic function was introduced in a series of three papers by Pierre François Verhulst between 1838 and 1847, who devised it as a model of population growth by adjusting the exponential growth model, under the guidance of Adolphe Quetelet. Verhulst first devised the function in the mid 1830s, publishing a brief note in 1838, then presented an expanded analysis and named the function in 1844 (published 1845); the third paper adjusted the correction term in his model of Belgian population growth.

The initial stage of growth is approximately exponential (geometric); then, as saturation begins, the growth slows to linear (arithmetic), and at maturity, growth stops. Verhulst did not explain the choice of the term "logistic" (French: logistique), but it is presumably in contrast to the logarithmic curve, and by analogy with arithmetic and geometric. His growth model is preceded by a discussion of arithmetic growth and geometric growth (whose curve he calls a logarithmic curve, instead of the modern term exponential curve), and thus "logistic growth" is presumably named by analogy, logistic being from Ancient Greek: λογῐστῐκός, romanized: logistikós, a traditional division of Greek mathematics. The term is unrelated to the military and management term logistics, which is instead from French: logis "lodgings", though some believe the Greek term also influenced logistics; see Logistics § Origin for details.

Mathematical properties

The standard logistic function is the logistic function with parameters (k = 1, x₀ = 0, L = 1) which yields

${\displaystyle {\begin{aligned}f(x)&={\frac {1}{1+e^{-x}}}={\frac {e^{x}}{e^{x}+1}}={\tfrac {1}{2}}+{\tfrac {1}{2}}\tanh({\tfrac {x}{2}})\\\end{aligned}}}$

In practice, due to the nature of the exponential function e^−x, it is often sufficient to compute the standard logistic function for x over a small range of real numbers such as a range contained in [−6, +6] as it quickly converges very close to its saturation values of 0 and 1. 

The logistic function has the symmetry property that:

${\displaystyle 1-f(x)=f(-x).}$

Thus, ${\displaystyle x\mapsto f(x)-1/2}$ is an odd function. 

The logistic function is an offset and scaled hyperbolic tangent function

${\displaystyle f(x)={\tfrac {1}{2}}+{\tfrac {1}{2}}\tanh({\tfrac {x}{2}})}$

or

${\displaystyle \tanh(x)=2\,f(2x)-1}$.

This follows from

${\displaystyle {\begin{aligned}\tanh(x)&={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}={\frac {e^{x}\cdot \left(1-e^{-2x}\right)}{e^{x}\cdot \left(1+e^{-2x}\right)}}\\[6pt]&=f(2x)-{\frac {e^{-2x}}{1+e^{-2x}}}=f(2x)-{\frac {e^{-2x}+1-1}{1+e^{-2x}}}=2f(2x)-1.\end{aligned}}}$

Derivative

The standard logistic function has an easily calculated derivative:

${\displaystyle f(x)={\frac {1}{1+e^{-x}}}={\frac {e^{x}}{1+e^{x}}},}$

 

 

${\displaystyle {\frac {d}{dx}}f(x)={\frac {e^{x}\cdot (1+e^{x})-e^{x}\cdot e^{x}}{(1+e^{x})^{2}}}={\frac {e^{x}}{(1+e^{x})^{2}}}=f(x)(1-f(x)).}$

The derivative of the logistic function is an even function, that is,

Integral

Conversely, its antiderivative can be computed by the substitution ${\displaystyle u=1+e^{x}}$, since ${\displaystyle f(x)={\frac {e^{x}}{1+e^{x}}}={\frac {u'}{u}}}$, so (dropping the constant of integration):

${\displaystyle \int {\frac {e^{x}}{1+e^{x}}}\,dx=\int {\frac {1}{u}}\,du=\log u=\log(1+e^{x})}$

In artificial neural networks, this is known as the softplus function, and (with scaling) is a smooth approximation of the ramp function, just as the logistic function (with scaling) is a smooth approximation of the Heaviside step function.

Logistic differential equation

The standard logistic function is the solution of the simple first-order non-linear ordinary differential equation

${\displaystyle {\frac {d}{dx}}f(x)=f(x)(1-f(x))}$

with boundary condition f(0) = 1/2. This equation is the continuous version of the logistic map. 

The qualitative behavior is easily understood in terms of the phase line: the derivative is 0 when the function is 1; and the derivative is positive for f between 0 and 1, and negative for f above 1 or less than 0 (though negative populations do not generally accord with a physical model). This yields an unstable equilibrium at 0, and a stable equilibrium at 1, and thus for any function value greater than 0 and less than 1, it grows to 1. 

The logistic equation is a special case of the Bernoulli differential equation and has the following solution:

${\displaystyle f(x)={\frac {e^{x}}{e^{x}+C}}}$

Choosing the constant of integration ${\displaystyle C=1}$ gives the other well-known form of the definition of the logistic curve

${\displaystyle f(x)={\frac {e^{x}}{e^{x}+1}}={\frac {1}{1+e^{-x}}}}$

More quantitatively, as can be seen from the analytical solution, the logistic curve shows early exponential growth for negative argument, which slows to linear growth of slope 1/4 for an argument near 0, then approaches 1 with an exponentially decaying gap. 

The logistic function is the inverse of the natural logit function and so can be used to convert the logarithm of odds into a probability. In mathematical notation the logistic function is sometimes written as expit in the same form as logit. The conversion from the log-likelihood ratio of two alternatives also takes the form of a logistic curve.

The hyperbolic tangent relationship leads to another form for the logistic function's derivative:

${\displaystyle {\frac {d}{dx}}f(x)={\frac {1}{4}}\operatorname {sech} ^{2}\left({\tfrac {x}{2}}\right),}$

which ties the logistic function into the logistic distribution.

Rotational symmetry about (0, ½)

The sum of the logistic function and its reflection about the vertical axis, f (−x) is

${\displaystyle {\frac {1}{1+e^{-x}}}+{\frac {1}{1+e^{-(-x)}}}={\frac {(1+e^{x})+(1+e^{-x})}{(1+e^{-x})(1+e^{x})}}={\frac {2+e^{x}+e^{-x}}{1+e^{x}+e^{-x}+e^{x-x}}}={\frac {2+e^{x}+e^{-x}}{2+e^{x}+e^{-x}}}=1.}$

The logistic function is thus rotationally symmetrical about the point (0, 1/2).

Applications

In ecology: modeling population growth

Pierre-François Verhulst (1804–1849)

 

A typical application of the logistic equation is a common model of population growth (see also population dynamics), originally due to Pierre-François Verhulst in 1838, where the rate of reproduction is proportional to both the existing population and the amount of available resources, all else being equal. The Verhulst equation was published after Verhulst had read Thomas Malthus' An Essay on the Principle of Population. Verhulst derived his logistic equation to describe the self-limiting growth of a biological population. The equation was rediscovered in 1911 by A. G. McKendrick for the growth of bacteria in broth and experimentally tested using a technique for nonlinear parameter estimation. The equation is also sometimes called the Verhulst-Pearl equation following its rediscovery in 1920 by Raymond Pearl (1879–1940) and Lowell Reed (1888–1966) of the Johns Hopkins University. Another scientist, Alfred J. Lotka derived the equation again in 1925, calling it the law of population growth. 

Letting P represent population size (N is often used in ecology instead) and t represent time, this model is formalized by the differential equation:

${\displaystyle {\frac {dP}{dt}}=rP\cdot \left(1-{\frac {P}{K}}\right)}$,

where the constant r defines the growth rate and K is the carrying capacity. 

In the equation, the early, unimpeded growth rate is modeled by the first term +rP. The value of the rate r represents the proportional increase of the population P in one unit of time. Later, as the population grows, the modulus of the second term (which multiplied out is −rP²/K) becomes almost as large as the first, as some members of the population P interfere with each other by competing for some critical resource, such as food or living space. This antagonistic effect is called the bottleneck, and is modeled by the value of the parameter K. The competition diminishes the combined growth rate, until the value of P ceases to grow (this is called maturity of the population). The solution to the equation (with ${\displaystyle P_{0}}$ being the initial population) is

${\displaystyle P(t)={\frac {KP_{0}e^{rt}}{K+P_{0}\left(e^{rt}-1\right)}}={\frac {K}{1+\left({\frac {K-P_{0}}{P_{0}}}\right)e^{-rt}}}}$,

where

${\displaystyle \lim _{t\to \infty }P(t)=K}$.

Which is to say that K is the limiting value of P: the highest value that the population can reach given infinite time (or come close to reaching in finite time). It is important to stress that the carrying capacity is asymptotically reached independently of the initial value P(0) > 0, and also in the case that P(0) > K. 

In ecology, species are sometimes referred to as r-strategist or K-strategist depending upon the selective processes that have shaped their life history strategies. Choosing the variable dimensions so that n measures the population in units of carrying capacity, and τ measures time in units of 1/r, gives the dimensionless differential equation

${\displaystyle {\frac {dn}{d\tau }}=n(1-n)}$.

Time-varying carrying capacity

Since the environmental conditions influence the carrying capacity, as a consequence it can be time-varying: K(t) > 0, leading to the following mathematical model:

${\displaystyle {\frac {dP}{dt}}=rP\cdot \left(1-{\frac {P}{K(t)}}\right)}$

A particularly important case is that of carrying capacity that varies periodically with period T:

${\displaystyle K(t+T)=K(t)}$.

It can be shown that in such a case, independently from the initial value P(0) > 0, P(t) will tend to a unique periodic solution P_*(t), whose period is T. 

A typical value of T is one year: In such case K(t) may reflect periodical variations of weather conditions.

Another interesting generalization is to consider that the carrying capacity K(t) is a function of the population at an earlier time, capturing a delay in the way population modifies its environment. This leads to a logistic delay equation, which has a very rich behavior, with bistability in some parameter range, as well as a monotonic decay to zero, smooth exponential growth, punctuated unlimited growth (i.e., multiple S-shapes), punctuated growth or alternation to a stationary level, oscillatory approach to a stationary level, sustainable oscillations, finite-time singularities as well as finite-time death.

In statistics and machine learning

Logistic functions are used in several roles in statistics. For example, they are the cumulative distribution function of the logistic family of distributions, and they are, a bit simplified, used to model the chance a chess player has to beat his opponent in the Elo rating system. More specific examples now follow.

Logistic regression

Logistic functions are used in logistic regression to model how the probability p of an event may be affected by one or more explanatory variables: an example would be to have the model

${\displaystyle p=f(a+bx)}$

where x is the explanatory variable and a and b are model parameters to be fitted and f is the standard logistic function. 

Logistic regression and other log-linear models are also commonly used in machine learning. A generalisation of the logistic function to multiple inputs is the softmax activation function, used in multinomial logistic regression. 

Another application of the logistic function is in the Rasch model, used in item response theory. In particular, the Rasch model forms a basis for maximum likelihood estimation of the locations of objects or persons on a continuum, based on collections of categorical data, for example the abilities of persons on a continuum based on responses that have been categorized as correct and incorrect.

Neural networks

Logistic functions are often used in neural networks to introduce nonlinearity in the model or to clamp signals to within a specified range. A popular neural net element computes a linear combination of its input signals, and applies a bounded logistic function to the result; this model can be seen as a "smoothed" variant of the classical threshold neuron. A common choice for the activation or "squashing" functions, used to clip for large magnitudes to keep the response of the neural network bounded is

${\displaystyle g(h)={\frac {1}{1+e^{-2\beta h}}}}$

which is a logistic function. These relationships result in simplified implementations of artificial neural networks with artificial neurons. Practitioners caution that sigmoidal functions which are antisymmetric about the origin (e.g. the hyperbolic tangent) lead to faster convergence when training networks with backpropagation.

The logistic function is itself the derivative of another proposed activation function, the softplus.

In medicine: modeling of growth of tumors

Another application of logistic curve is in medicine, where the logistic differential equation is used to model the growth of tumors. This application can be considered an extension of the above-mentioned use in the framework of ecology (see also the Generalized logistic curve, allowing for more parameters). Denoting with X(t) the size of the tumor at time t, its dynamics are governed by:

${\displaystyle X'=r\left(1-{\frac {X}{K}}\right)X}$

which is of the type:

${\displaystyle X'=F(X)X,\qquad F'(X)\leq 0}$

where F(X) is the proliferation rate of the tumor. 

If a chemotherapy is started with a log-kill effect, the equation may be revised to be

${\displaystyle X'=r\left(1-{\frac {X}{K}}\right)X-c(t)X,}$

where c(t) is the therapy-induced death rate. In the idealized case of very long therapy, c(t) can be modeled as a periodic function (of period T) or (in case of continuous infusion therapy) as a constant function, and one has that

${\displaystyle {\frac {1}{T}}\int _{0}^{T}c(t)\,dt>r\to \lim _{t\to +\infty }x(t)=0}$

i.e. if the average therapy-induced death rate is greater than the baseline proliferation rate then there is the eradication of the disease. Of course, this is an oversimplified model of both the growth and the therapy (e.g. it does not take into account the phenomenon of clonal resistance).

In chemistry: reaction models

The concentration of reactants and products in autocatalytic reactions follow the logistic function. The degradation of platinum group metal-free (PGM-free) oxygen reduction reaction (ORR) catalyst in fuel cell cathodes follows the logistic decay function, suggesting an autocatalytic degradation mechanism.

In physics: Fermi distribution

The logistic function determines the statistical distribution of fermions over the energy states of a system in thermal equilibrium. In particular, it is the distribution of the probabilities that each possible energy level is occupied by a fermion, according to Fermi–Dirac statistics.

In linguistics: language change

In linguistics, the logistic function can be used to model language change: an innovation that is at first marginal begins to spread more quickly with time, and then more slowly as it becomes more universally adopted.

In agriculture: modeling crop response

The logistic S-curve can be used for modeling the crop response to changes in growth factors. There are two types of response functions: positive and negative growth curves. For example, the crop yield may increase with increasing value of the growth factor up to a certain level (positive function), or it may decrease with increasing growth factor values (negative function owing to a negative growth factor), which situation requires an inverted S-curve. 

S-curve model for yield versus depth of watertable.

 

Inverted S-curve model for yield versus soil salinity.

In economics and sociology: diffusion of innovations

The logistic function can be used to illustrate the progress of the diffusion of an innovation through its life cycle. 

In The Laws of Imitation (1890), Gabriel Tarde describes the rise and spread of new ideas through imitative chains. In particular, Tarde identifies three main stages through which innovations spread: the first one corresponds to the difficult beginnings, during which the idea has to struggle within a hostile environment full of opposing habits and beliefs; the second one corresponds to the properly exponential take-off of the idea, with ${\displaystyle f(x)=2^{x}}$; finally, the third stage is logarithmic, with ${\displaystyle f(x)=\log(x)}$, and corresponds to the time when the impulse of the idea gradually slows down while, simultaneously new opponent ideas appear. The ensuing situation halts or stabilizes the progress of the innovation, which approaches an asymptote.

In the history of economy, when new products are introduced there is an intense amount of research and development which leads to dramatic improvements in quality and reductions in cost. This leads to a period of rapid industry growth. Some of the more famous examples are: railroads, incandescent light bulbs, electrification, cars and air travel. Eventually, dramatic improvement and cost reduction opportunities are exhausted, the product or process are in widespread use with few remaining potential new customers, and markets become saturated. 

Logistic analysis was used in papers by several researchers at the International Institute of Applied Systems Analysis (IIASA). These papers deal with the diffusion of various innovations, infrastructures and energy source substitutions and the role of work in the economy as well as with the long economic cycle. Long economic cycles were investigated by Robert Ayres (1989). Cesare Marchetti published on long economic cycles and on diffusion of innovations. Arnulf Grübler's book (1990) gives a detailed account of the diffusion of infrastructures including canals, railroads, highways and airlines, showing that their diffusion followed logistic shaped curves.

Carlota Perez used a logistic curve to illustrate the long (Kondratiev) business cycle with the following labels: beginning of a technological era as irruption, the ascent as frenzy, the rapid build out as synergy and the completion as maturity.

Dyscalculia

From Wikipedia, the free encyclopedia

Dyscalculia
Pronunciation	/ˌdɪskælˈkjuːliə/
Specialty	Pediatrics
Duration	Lifetime

Dyscalculia /ˌdɪskælˈkjuːliə/ is difficulty in learning or comprehending arithmetic, such as difficulty in understanding numbers, learning how to manipulate numbers, performing mathematical calculations and learning facts in mathematics. It is generally seen as the mathematical equivalent to dyslexia.

It can occur in people from across the whole IQ range, along with difficulties with time, measurement, and spatial reasoning. Estimates of the prevalence of dyscalculia range between 3 and 6% of the population. In 2015, it was established that 11% of children with dyscalculia also have ADHD. Dyscalculia has also been associated with people who have Turner syndrome and people who have spina bifida.

Mathematical disabilities can occur as the result of some types of brain injury, in which case the proper term, acalculia, is to distinguish it from dyscalculia which is of innate, genetic or developmental origin.

Signs and symptoms

The earliest appearance of dyscalculia is typically a deficit in the ability to know, from a brief glance and without counting, how many objects there are in a small group. Children as young as 5 can subitize 6 objects, especially looking at a die. However, children with dyscalculia can subitize fewer objects and even when correct take longer to identify the number than their age-matched peers. Dyscalculia often looks different at different ages. It tends to become more apparent as kids get older; however, symptoms can appear as early as preschool. Common symptoms of dyscalculia are, having difficulty with mental math, trouble analyzing time and reading an analog clock, struggle with motor sequencing that involves numbers, and often they will count on their fingers when adding numbers.

Common symptoms

Dyscalculia is characterized by difficulties with common arithmetic tasks. These difficulties may include:

• Difficulty reading analog clocks
• Difficulty stating which of two numbers is larger
• Sequencing issues
• Inability to comprehend financial planning or budgeting, sometimes even at a basic level; for example, estimating the cost of the items in a shopping basket or balancing a checkbook
• Inconsistent results in addition, subtraction, multiplication and division
• Visualizing numbers as meaningless or nonsensical symbols, rather than perceiving them as characters indicating a numerical value (hence the misnomer, "math dyslexia")
• Difficulty with multiplication, subtraction, addition, and division tables, mental arithmetic, etc.
• Problems with differentiating between left and right
• A "warped" sense of spatial awareness, or an understanding of shapes, distance, or volume that seems more like guesswork than actual comprehension
• Difficulty with time, directions, recalling schedules, sequences of events, keeping track of time, frequently late or early
• Poor memory (retention and retrieval) of math concepts; may be able to perform math operations one day, but draw a blank the next; may be able to do book work but then fails tests
• Ability to grasp math on a conceptual level, but an inability to put those concepts into practice
• Difficulty recalling the names of numbers, or thinking that certain different numbers "feel" the same (e.g. frequently interchanging the same two numbers for each other when reading or recalling them)
• Difficulty reading musical notation
• Difficulty with choreographed dance steps
• Difficulty working backwards in time (e.g. What time to leave if needing to be somewhere at 'X' time)
• Having particular difficulty mentally estimating the measurement of an object or distance (e.g., whether something is 3 or 6 meters (10 or 20 feet) away)
• When writing, reading and recalling numbers, mistakes may occur in the areas such as: number additions, substitutions, transpositions, omissions, and reversals
• Inability to grasp and remember mathematical concepts, rules, formulae, and sequences
• Inability to concentrate on mentally intensive tasks
• Mistaken recollection of names, poor name/face retrieval, may substitute names beginning with same letter.

Persistence in children

Although many researchers believe dyscalculia to be a persistent disorder, evidence on the persistence of dyscalculia remains mixed. For instance, in a study done by Mazzocco and Myers (2003), researchers evaluated children on a slew of measures and selected their most consistent measure as their best diagnostic criterion: a stringent 10th-percentile cut-off on the TEMA-2. Even with their best criterion, they found dyscalculia diagnoses for children longitudinally did not persist; only 65% of students who were ever diagnosed over the course of four years were diagnosed for at least two years. The percentage of children who were diagnosed in two consecutive years was further reduced. It is unclear whether this was the result of misdiagnosed children improving in mathematics and spatial awareness as they progressed as normal, or that the subjects who showed improvement were accurately diagnosed, but exhibited signs of a non-persistent learning disability.

Persistence in adults

There are very few studies of adults with dyscalculia who have had a history of it growing up, but such studies have shown that it can persist into adulthood. It can affect major parts of an adult's life. Most adults with dyscalculia have a hard time processing math at a 4th grade level. For 1st-4th grade level, many adults will know what to do for the math problem, but they will often get them wrong because of "careless errors," although they are not careless when it comes to the problem. The adults cannot process their errors on the math problems or may not even recognize that they have made these errors. Visual-spatial input, auditory input, and touch input will be affected due to these processing errors. Dyscalculics may have a difficult time adding numbers in a column format because their mind can mix up the numbers, and it is possible that they may get the same answer twice due to their mind processing the problem incorrectly. Dyscalculics can have problems determining differences in different coins and their size or giving the correct amount of change and if numbers are grouped together, it is possible that they cannot determine which has less or more. If a dyscalculic is asked to choose the greater of two numbers, with the lesser number in a larger font than the greater number, they may take the question literally and pick the number with the bigger font. Adults with dyscalculia have a tough time with directions while driving and with controlling their finances, which causes difficulties on a day-to-day basis.

College students or other adult learners

College students particularly may have a tougher time due to the fast pace and change in difficulty of the work they are given. As a result of this, students may develop a lot of anxiety and frustration. After dealing with their anxiety for a long time, students can become averse to math and try to avoid it as much as possible, which may result in lower grades in math courses. However, students with dyscalculia often do exceptionally in writing, reading, and speaking. Students may try to succeed through determination and persistence because of their inability to do well with numbers. They may try to keep a positive attitude even with the frustration and anxiety because they want to meet their goal in life. The problem, when it comes to college, is that professors cannot grade entirely on their persistence, determination, and efforts. Students need to figure out ways to overcome their difficulties. There are a lot of services that schools can provide for students. In the 21st century there is evidence that there will be an increase in enrollment for students with learning disabilities in community colleges.

Causes

Both domain-general and domain-specific causes have been put forth. With respect to pure developmental dyscalculia, domain-general causes are unlikely as they should not impair one’s ability in the numerical domain without also affecting other domains such as reading. 

Two competing domain-specific hypotheses about the causes of developmental dyscalculia have been proposed – the magnitude representation (or number module deficit hypothesis) and the access deficit hypothesis.

Magnitude representation deficit

Dehaene's "number sense" theory suggests that approximate numerosities are automatically ordered in an ascending manner on a mental number line. The mechanism to represent and process non-symbolic magnitude (e.g., number of dots) is often known as the "approximate number system" (ANS), and a core deficit in the precision of the ANS, known as the "magnitude representation hypothesis" or "number module deficit hypothesis", has been proposed as an underlying cause of developmental dyscalculia.

In particular, the structural features of the ANS is theoretically supported by a phenomenon called the "numerical distance effect", which has been robustly observed in numerical comparison tasks. Typically developing individuals are less accurate and slower in comparing pairs of numbers closer together (e.g., 7 and 8) than further apart (e.g., 2 and 9). A related "numerical ratio effect" (in which the ratio between two numbers varies but the distance is kept constant, e.g., 2 vs. 5 and 4 vs. 7) based on the Weber's law has also been used to further support the structure of the ANS. The numerical ratio effect is observed when individuals are less accurate and slower in comparing pairs of numbers that have a larger ratio (e.g., 8 and 9, ratio = 8/9) than a smaller ratio (2 and 3; ratio = 2/3). A larger numerical distance or ratio effect with comparison of sets of objects (i.e., non-symbolic) is thought to reflect a less precise ANS, and the ANS acuity has been found to correlate with math achievement in typically developing children and also in adults.

More importantly, several behavioral studies have found that children with developmental dyscalculia show an attenuated distance/ratio effect than typically developing children. Moreover, neuroimaging studies have also provided additional insights even when behavioral difference in distance/ratio effect might not be clearly evident. For example, Gavin R. Price and colleagues found that children with developmental dyscalculia showed no differential distance effect on reaction time relative to typically developing children, but they did show a greater effect of distance on response accuracy. They also found that the right intraparietal sulcus in children with developmental dyscalculia was not modulated to the same extent in response to non-symbolic numerical processing as in typically developing children. With the robust implication of the intraparietal sulcus in magnitude representation, it is possible that children with developmental dyscalculia have a weak magnitude representation in the parietal region. Yet, it does not rule out an impaired ability to access and manipulate numerical quantities from their symbolic representations (e.g., Arabic digits). 

This shows the part of the brain where the sulcus is located in the parietal lobe.

 

Moreover, findings from a cross-sectional study suggest that children with developmental dyscalculia might have a delayed development in their numerical magnitude representation by as much as five years. However, the lack of longitudinal studies still leaves the question open as to whether the deficient numerical magnitude representation is a delayed development or impairment.

Access deficit hypothesis

Rousselle & Noël propose that dyscalculia is caused by the inability to map preexisting representations of numerical magnitude onto symbolic Arabic digits. Evidence for this hypothesis is based on research studies that have found that individuals with dyscalculia are proficient on tasks that measure knowledge of non-symbolic numerical magnitude (i.e., non-symbolic comparison tasks) but show an impaired ability to process symbolic representations of number (i.e., symbolic comparison tasks). Neuroimaging studies also report increased activation in the right intraparietal sulcus during tasks that measure symbolic but not non-symbolic processing of numerical magnitude. However, support for the access deficit hypothesis is not consistent across research studies.

Diagnosis

At its most basic level, dyscalculia is a learning disability affecting the normal development of arithmetic skills.

A consensus has not yet been reached on appropriate diagnostic criteria for dyscalculia. Mathematics is a specific domain that is complex (i.e. includes many different processes, such as arithmetic, algebra, word problems, geometry, etc.) and cumulative (i.e. the processes build on each other such that mastery of an advanced skill requires mastery of many basic skills). Thus dyscalculia can be diagnosed using different criteria, and frequently is; this variety in diagnostic criteria leads to variability in identified samples, and thus variability in research findings regarding dyscalculia.

The example of each condition in the numerical stroop effect task

 

Other than using achievement tests as diagnostic criteria, researchers often rely on domain-specific tests (i.e. tests of working memory, executive function, inhibition, intelligence, etc.) and teacher evaluations to create a more comprehensive diagnosis. Alternatively, fMRI research has shown that the brains of the neurotypical children can be reliably distinguished from the brains of the dyscalculic children based on the activation in the prefrontal cortex. However, due to the cost and time limitations associated with brain and neural research, these methods will likely not be incorporated into diagnostic criteria despite their effectiveness.

Types

Research on subtypes of dyscalculia has begun without consensus; preliminary research has focused on comorbid learning disorders as subtyping candidates. The most common comorbidity in individuals with dyscalculia is dyslexia. Most studies done with comorbid samples versus dyscalculic-only samples have shown different mechanisms at work and additive effects of comorbidity, indicating that such subtyping may not be helpful in diagnosing dyscalculia. But there is variability in results at present.

Due to high comorbidity with other disabilities such as dyslexia and ADHD, some researchers have suggested the possibility of subtypes of mathematical disabilities with different underlying profiles and causes. Whether a particular subtype is specifically termed "dyscalculia" as opposed to a more general mathematical learning disability is somewhat under debate in the scientific literature.

• Semantic memory: This subtype often coexists with reading disabilities such as dyslexia and is characterized by poor representation and retrieval from long-term memory. These processes share a common neural pathway in the left angular gyrus, which has been shown to be selective in arithmetic fact retrieval strategies and symbolic magnitude judgments. This region also shows low functional connectivity with language-related areas during phonological processing in adults with dyslexia. Thus, disruption to the left angular gyrus can cause both reading impairments and difficulties in calculation. This has been observed in individuals with Gerstmann syndrome, of which dyscalculia is one of constellation of symptoms.
• Procedural concepts: Research by Geary has shown that in addition to increased problems with fact retrieval, children with math disabilities may rely on immature computational strategies. Specifically, children with mathematical disabilities showed poor command of counting strategies unrelated to their ability to retrieve numeric facts. This research notes that it is difficult to discern whether poor conceptual knowledge is indicative of a qualitative deficit in number processing or simply a delay in typical mathematical development.
• Working memory: Studies have found that children with dyscalculia showed impaired performance on working memory tasks compared to neurotypical children. Furthermore, research has shown that children with dyscalculia have weaker activation of the intraparietal sulcus during visuospatial working memory tasks. Brain activity in this region during such tasks has been linked to overall arithmetic performance, indicating that numerical and working memory functions may converge in the intraparietal sulcus. However, working memory problems are confounded with domain-general learning difficulties, thus these deficits may not be specific to dyscalculia but rather may reflect a greater learning deficit. Dysfunction in prefrontal regions may also lead to deficits in working memory and other executive function, accounting for comorbidity with ADHD.

Studies have also shown indications of causes due to congenital or hereditary disorders, but evidence of this is not yet concrete.

Treatment

To date, very few interventions have been developed specifically for individuals with dyscalculia. Concrete manipulation activities have been used for decades to train basic number concepts for remediation purposes. This method facilitates the intrinsic relationship between a goal, the learner’s action, and the informational feedback on the action. A one-to-one tutoring paradigm designed by Lynn Fuchs and colleagues which teaches concepts in arithmetic, number concepts, counting, and number families using games, flash cards, and manipulables has proven successful in children with generalized math learning difficulties, but intervention has yet to be tested specifically on children with dyscalculia. These methods require specially trained teachers working directly with small groups or individual students. As such, instruction time in the classroom is necessarily limited. For this reason, several research groups have developed computer adaptive training programs designed to target deficits unique to dyscalculic individuals. 

Software intended to remediate dyscalculia has been developed. While computer adaptive training programs are modeled after one-to-one type interventions, they provide several advantages. Most notably, individuals are able to practice more with a digital intervention than is typically possible with a class or teacher. As with one-to-one interventions, several digital interventions have also proven successful in children with generalized math learning difficulties. Räsänen and colleagues have found that games such as The Number Race and Graphogame-math can improve performance on number comparison tasks in children with generalized math learning difficulties.

Several digital interventions have been developed for dyscalculics specifically. Each attempts to target basic processes that are associated with maths difficulties. Rescue Calcularis was one early computerized intervention that sought to improve the integrity of and access to the mental number line. Other digital interventions for dyscalculia adapt games, flash cards, and manipulables to function through technology.

While each intervention claims to improve basic numerosity skills, the authors of these interventions do admit that repetition and practice effects may be a factor involved in reported performance gains. An additional criticism is that these digital interventions lack the option to manipulate numerical quantities. While the previous two games provide the correct answer, the individual using the intervention cannot actively determine, through manipulation, what the correct answer should be. Butterworth and colleagues argued that games like The Number Bonds, which allows an individual to compare different sized rods, should be the direction that digital interventions move towards. Such games use manipulation activities to provide intrinsic motivation towards content guided by dyscalculia research. One of these serious games is Meister Cody – Talasia, an online training that includes the CODY Assessment – a diagnostic test for detecting dyscalculia. Based on these findings, Rescue Calcluaris was extended by adaptation algorithms and game forms allowing manipulation by the learners. It was found to improve addition, subtraction and number line tasks, and was made available as Dybuster Calcularis.

A study used transcranial direct current stimulation (TDCS) to the parietal lobe during numerical learning and demonstrated selective improvement of numerical abilities that was still present six months later in typically developing individuals. Improvement were achieved by applying anodal current to the right parietal lobe and cathodal current to the left parietal lobe and contrasting it with the reverse setup. When the same research group used tDCS in a training study with two dyscalculic individuals, the reverse setup (left anodal, right cathodal) demonstrated improvement of numerical abilities.

Epidemiology

Dyscalculia is thought to be present in 3–6% of the general population, but estimates by country and sample vary somewhat. Many studies have found prevalence rates by gender to be equivalent. Those that find gender difference in prevalence rates often find dyscalculia higher in females, but some few studies have found prevalence rates higher in males.

History

The term 'dyscalculia' was coined in the 1940s, but it was not completely recognized until 1974 by the work of Czechoslovakian researcher Ladislav Kosc. Kosc defined dyscalculia as "a structural disorder of mathematical abilities." His research proved that the learning disability was caused by impairments to certain parts of the brain that control mathematical calculations and not because symptomatic individuals were 'mentally handicapped'. Researchers now sometimes use the terms “math dyslexia” or “math learning disability” when they mention the condition. Cognitive disabilities specific to mathematics were originally identified in case studies with patients who experienced specific arithmetic disabilities as a result of damage to specific regions of the brain. More commonly, dyscalculia occurs developmentally as a genetically linked learning disability which affects a person's ability to understand, remember, or manipulate numbers or number facts (e.g., the multiplication tables). The term is often used to refer specifically to the inability to perform arithmetic operations, but is also defined by some educational professionals and cognitive psychologists such as Stanislas Dehaene and Brian Butterworth as a more fundamental inability to conceptualize numbers as abstract concepts of comparative quantities (a deficit in "number sense"), which these researchers consider to be a foundational skill upon which other mathematics abilities build. Symptoms of dyscalculia include the delay of simple counting, inability to memorize simple arithmetic facts such as adding, subtracting, etc. There are few known symptoms because little research has been done on the topic.

Etymology

The term dyscalculia dates back to at least 1949.

Dyscalculia comes from Greek and Latin and means "counting badly". The prefix "dys-" comes from Greek and means "badly". The root "calculia" comes from the Latin "calculare", which means "to count" and which is also related to "calculation" and "calculus".