HOME

TheInfoList



OR:

Models of neural computation are attempts to elucidate, in an abstract and mathematical fashion, the core principles that underlie information processing in biological nervous systems, or functional components thereof. This article aims to provide an overview of the most definitive models of neuro-biological computation as well as the tools commonly used to construct and analyze them.


Introduction

Due to the complexity of nervous system behavior, the associated experimental error bounds are ill-defined, but the relative merit of the different
models A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...
of a particular subsystem can be compared according to how closely they reproduce real-world behaviors or respond to specific input signals. In the closely related field of computational
neuroethology Neuroethology is the evolutionary and comparative approach to the study of animal behavior and its underlying mechanistic control by the nervous system. It is an interdisciplinary science that combines both neuroscience (study of the nervous syste ...
, the practice is to include the environment in the model in such a way that the loop is closed. In the cases where competing models are unavailable, or where only gross responses have been measured or quantified, a clearly formulated model can guide the scientist in designing experiments to probe biochemical mechanisms or network connectivity. In all but the simplest cases, the mathematical equations that form the basis of a model cannot be solved exactly. Nevertheless, computer technology, sometimes in the form of specialized software or hardware architectures, allow scientists to perform iterative calculations and search for plausible solutions. A computer chip or a robot that can interact with the natural environment in ways akin to the original organism is one embodiment of a useful model. The ultimate measure of success is however the ability to make testable predictions.


General criteria for evaluating models


Speed of information processing

The rate of information processing in biological neural systems are constrained by the speed at which an action potential can propagate down a nerve fibre. This conduction velocity ranges from 1 m/s to over 100 m/s, and generally increases with the diameter of the neuronal process. Slow in the timescales of biologically-relevant events dictated by the speed of sound or the force of gravity, the nervous system overwhelmingly prefers parallel computations over serial ones in time-critical applications.


Robustness

A model is robust if it continues to produce the same computational results under variations in inputs or operating parameters introduced by noise. For example, the direction of motion as computed by a robust
motion detector A motion detector is an electrical device that utilizes a sensor to detect nearby motion. Such a device is often integrated as a component of a system that automatically performs a task or alerts a user of motion in an area. They form a vital c ...
would not change under small changes of
luminance Luminance is a photometric measure of the luminous intensity per unit area of light travelling in a given direction. It describes the amount of light that passes through, is emitted from, or is reflected from a particular area, and falls with ...
, contrast or velocity jitter. For simple mathematical models of neuron, for example the dependence of spike patterns on signal delay is much weaker than the dependence on changes in "weights" of interneuronal connections.


Gain control

This refers to the principle that the response of a nervous system should stay within certain bounds even as the inputs from the environment change drastically. For example, when adjusting between a sunny day and a moonless night, the retina changes the relationship between light level and neuronal output by a factor of more than 10^6 so that the signals sent to later stages of the visual system always remain within a much narrower range of amplitudes.


Linearity versus nonlinearity

A linear system is one whose response in a specified unit of measure, to a set of inputs considered at once, is the sum of its responses due to the inputs considered individually.
Linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
systems are easier to analyze mathematically and are a persuasive assumption in many models including the McCulloch and Pitts neuron, population coding models, and the simple neurons often used in
Artificial neural network Artificial neural networks (ANNs), usually simply called neural networks (NNs) or neural nets, are computing systems inspired by the biological neural networks that constitute animal brains. An ANN is based on a collection of connected unit ...
s. Linearity may occur in the basic elements of a neural circuit such as the response of a postsynaptic neuron, or as an emergent property of a combination of nonlinear subcircuits. Though linearity is often seen as incorrect, there has been recent work suggesting it may, in fact, be biophysically plausible in some cases.


Examples

A computational neural model may be constrained to the level of biochemical signalling in individual
neurons A neuron, neurone, or nerve cell is an electrically excitable cell that communicates with other cells via specialized connections called synapses. The neuron is the main component of nervous tissue in all animals except sponges and placozoa ...
or it may describe an entire organism in its environment. The examples here are grouped according to their scope.


Models of information transfer in neurons

The most widely used models of information transfer in biological neurons are based on analogies with electrical circuits. The equations to be solved are time-dependent differential equations with electro-dynamical variables such as current, conductance or resistance, capacitance and voltage.


Hodgkin–Huxley model and its derivatives

The Hodgkin–Huxley model, widely regarded as one of the great achievements of 20th-century biophysics, describes how
action potential An action potential occurs when the membrane potential of a specific cell location rapidly rises and falls. This depolarization then causes adjacent locations to similarly depolarize. Action potentials occur in several types of animal cells ...
s in neurons are initiated and propagated in axons via
voltage-gated ion channel Voltage-gated ion channels are a class of transmembrane proteins that form ion channels that are activated by changes in the electrical membrane potential near the channel. The membrane potential alters the conformation of the channel proteins, ...
s. It is a set of
nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many oth ...
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
s that were introduced by
Alan Lloyd Hodgkin Sir Alan Lloyd Hodgkin (5 February 1914 – 20 December 1998) was an English physiologist and biophysicist who shared the 1963 Nobel Prize in Physiology or Medicine with Andrew Huxley and John Eccles. Early life and education Hodgkin was ...
and
Andrew Huxley Sir Andrew Fielding Huxley (22 November 191730 May 2012) was an English physiologist and biophysicist. He was born into the prominent Huxley family. After leaving Westminster School in central London, he went to Trinity College, Cambridge ...
in 1952 to explain the results of
voltage clamp The voltage clamp is an experimental method used by electrophysiologists to measure the ion currents through the membranes of excitable cells, such as neurons, while holding the membrane voltage at a set level. A basic voltage clamp will itera ...
experiments on the
squid giant axon The squid giant axon is the very large (up to 1.5 mm in diameter; typically around 0.5 mm) axon that controls part of the water jet propulsion system in squid. It was first described by L. W. Williams in 1909, but this discovery was f ...
. Analytic solutions do not exist, but the Levenberg–Marquardt algorithm, a modified
Gauss–Newton algorithm The Gauss–Newton algorithm is used to solve non-linear least squares problems, which is equivalent to minimizing a sum of squared function values. It is an extension of Newton's method for finding a minimum of a non-linear function. Since a sum ...
, is often used to fit these equations to voltage-clamp data. The FitzHugh–Nagumo model is a simplication of the Hodgkin–Huxley model. The
Hindmarsh–Rose model The Hindmarsh–Rose model of neuronal activity is aimed to study the spiking-bursting behavior of the membrane potential observed in experiments made with a single neuron. The relevant variable is the membrane potential, ''x''(''t''), which is ...
is an extension which describes neuronal spike bursts. The Morris–Lecar model is a modification which does not generate spikes, but describes slow-wave propagation, which is implicated in the inhibitory synaptic mechanisms of
central pattern generator Central pattern generators (CPGs) are self-organizing biological neural circuits that produce rhythmic outputs in the absence of rhythmic input. They are the source of the tightly-coupled patterns of neural activity that drive rhythmic and stereo ...
s.


Solitons

The
soliton model The soliton hypothesis in neuroscience is a model that claims to explain how action potentials are initiated and conducted along axons based on a thermodynamic theory of nerve pulse propagation. It proposes that the signals travel along the ce ...
is an alternative to the Hodgkin–Huxley model that claims to explain how
action potentials An action potential occurs when the membrane potential of a specific cell location rapidly rises and falls. This depolarization then causes adjacent locations to similarly depolarize. Action potentials occur in several types of animal cells ...
are initiated and conducted in the form of certain kinds of
solitary Solitary is the state of being alone or in solitude. The term may refer to: * shortened form of solitary confinement * Solitary animal, an animal that does not live with others in its species * Solitary but social, a type of social organization ...
sound In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by ...
(or
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
) pulses that can be modeled as
soliton In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the me ...
s along
axon An axon (from Greek ἄξων ''áxōn'', axis), or nerve fiber (or nerve fibre: see spelling differences), is a long, slender projection of a nerve cell, or neuron, in vertebrates, that typically conducts electrical impulses known as action p ...
s, based on a thermodynamic theory of nerve pulse propagation.


Transfer functions and linear filters

This approach, influenced by
control theory Control theory is a field of mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive ...
and
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, treats neurons and synapses as time-invariant entities that produce outputs that are linear combinations of input signals, often depicted as sine waves with a well-defined temporal or spatial frequencies. The entire behavior of a neuron or synapse are encoded in a
transfer function In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that theoretically models the system's output for each possible input. They are widely used ...
, lack of knowledge concerning the exact underlying mechanism notwithstanding. This brings a highly developed mathematics to bear on the problem of information transfer. The accompanying taxonomy of
linear filter Linear filters process time-varying input signals to produce output signals, subject to the constraint of linearity. In most cases these linear filters are also time invariant (or shift invariant) in which case they can be analyzed exactly using ...
s turns out to be useful in characterizing neural circuitry. Both low- and
high-pass filter A high-pass filter (HPF) is an electronic filter that passes signals with a frequency higher than a certain cutoff frequency and attenuates signals with frequencies lower than the cutoff frequency. The amount of attenuation for each frequency ...
s are postulated to exist in some form in sensory systems, as they act to prevent information loss in high and low contrast environments, respectively. Indeed, measurements of the transfer functions of neurons in the
horseshoe crab Horseshoe crabs are marine and brackish water arthropods of the family Limulidae and the only living members of the order Xiphosura. Despite their name, they are not true crabs or crustaceans: they are chelicerates, most closely related to ar ...
retina according to linear systems analysis show that they remove short-term fluctuations in input signals leaving only the long-term trends, in the manner of low-pass filters. These animals are unable to see low-contrast objects without the help of optical distortions caused by underwater currents.


Models of computations in sensory systems


Lateral inhibition in the retina: Hartline–Ratliff equations

In the retina, an excited neural receptor can suppress the activity of surrounding neurons within an area called the inhibitory field. This effect, known as
lateral inhibition In neurobiology, lateral inhibition is the capacity of an excited neuron to reduce the activity of its neighbors. Lateral inhibition disables the spreading of action potentials from excited neurons to neighboring neurons in the lateral direction ...
, increases the contrast and sharpness in visual response, but leads to the epiphenomenon of Mach bands. This is often illustrated by the
optical illusion Within visual perception, an optical illusion (also called a visual illusion) is an illusion caused by the visual system and characterized by a visual perception, percept that arguably appears to differ from reality. Illusions come in a wide v ...
of light or dark stripes next to a sharp boundary between two regions in an image of different luminance. The Hartline-Ratliff model describes interactions within a group of ''p''
photoreceptor cell A photoreceptor cell is a specialized type of neuroepithelial cell found in the retina that is capable of visual phototransduction. The great biological importance of photoreceptors is that they convert light (visible electromagnetic radiat ...
s. Assuming these interactions to be linear, they proposed the following relationship for the steady-state response rate r_p of the given ''p''-th photoreceptor in terms of the steady-state response rates r_j of the ''j'' surrounding receptors: r_=\left, \left r_-r_^\\right. Here, e_p is the excitation of the target ''p''-th receptor from sensory transduction r_^o is the associated threshold of the firing cell, and k_ is the coefficient of inhibitory interaction between the ''p''-th and the ''j''th receptor. The inhibitory interaction decreases with distance from the target ''p''-th receptor.


Cross-correlation in sound localization: Jeffress model

According to Jeffress, in order to compute the location of a sound source in space from interaural time differences, an auditory system relies on delay lines: the induced signal from an
ipsilateral Standard anatomical terms of location are used to unambiguously describe the anatomy of animals, including humans. The terms, typically derived from Latin or Greek roots, describe something in its standard anatomical position. This position prov ...
auditory receptor to a particular neuron is delayed for the same time as it takes for the original sound to go in space from that ear to the other. Each postsynaptic cell is differently delayed and thus specific for a particular inter-aural time difference. This theory is equivalent to the mathematical procedure of
cross-correlation In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
. Following Fischer and Anderson, the response of the postsynaptic neuron to the signals from the left and right ears is given by y_\left(t\right) - y_\left(t\right) where y_\left(t\right)=\int_^u_\left(\sigma\right)w\left(t-\sigma\right)d\sigma y_\left(t\right)=\int_^u_\left(\sigma\right)w\left(t-\sigma\right)d\sigma and w\left(t-\sigma\right) represents the delay function. This is not entirely correct and a clear eye is needed to put the symbols in order. Structures have been located in the barn owl which are consistent with Jeffress-type mechanisms.


Cross-correlation for motion detection: Hassenstein–Reichardt model

A motion detector needs to satisfy three general requirements: pair-inputs, asymmetry and nonlinearity. The cross-correlation operation implemented asymmetrically on the responses from a pair of photoreceptors satisfies these minimal criteria, and furthermore, predicts features which have been observed in the response of neurons of the lobula plate in bi-wing insects. The master equation for response is R = A_1(t-\tau)B_2(t) - A_2(t - \tau)B_1(t) The HR model predicts a peaking of the response at a particular input temporal frequency. The conceptually similar Barlow–Levick model is deficient in the sense that a stimulus presented to only one receptor of the pair is sufficient to generate a response. This is unlike the HR model, which requires two correlated signals delivered in a time ordered fashion. However the HR model does not show a saturation of response at high contrasts, which is observed in experiment. Extensions of the Barlow-Levick model can provide for this discrepancy.


Watson–Ahumada model for motion estimation in humans

This uses a cross-correlation in both the spatial and temporal directions, and is related to the concept of optical flow.


Anti-Hebbian adaptation: spike-timing dependent plasticity

* *


Models of

sensory-motor coupling Sensory-motor coupling is the coupling or integration of the sensory system and motor system. Sensorimotor integration is not a static process. For a given stimulus, there is no one single motor command. "Neural responses at almost every stage of a ...


Neurophysiological metronomes: neural circuits for pattern generation

Mutually inhibitory processes are a unifying motif of all
central pattern generator Central pattern generators (CPGs) are self-organizing biological neural circuits that produce rhythmic outputs in the absence of rhythmic input. They are the source of the tightly-coupled patterns of neural activity that drive rhythmic and stereo ...
s. This has been demonstrated in the stomatogastric (STG) nervous system of crayfish and lobsters. Two and three-cell oscillating networks based on the STG have been constructed which are amenable to mathematical analysis, and which depend in a simple way on synaptic strengths and overall activity, presumably the knobs on these things. The mathematics involved is the theory of
dynamical systems In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a ...
.


Feedback and control: models of flight control in the fly

Flight control in the fly is believed to be mediated by inputs from the visual system and also the
halteres ''Halteres'' (; singular ''halter'' or ''haltere'') (from grc, ἁλτῆρες, weights held in the hands to give an impetus in leaping) are a pair of small club-shaped organs on the body of two orders of flying insects that provide infor ...
, a pair of knob-like organs which measure angular velocity. Integrated computer models of ''
Drosophila ''Drosophila'' () is a genus of flies, belonging to the family Drosophilidae, whose members are often called "small fruit flies" or (less frequently) pomace flies, vinegar flies, or wine flies, a reference to the characteristic of many speci ...
'', short on neuronal circuitry but based on the general guidelines given by
control theory Control theory is a field of mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive ...
and data from the tethered flights of flies, have been constructed to investigate the details of flight control.


Cerebellum sensory motor control

Tensor network theory Tensor network theory is a theory of brain function (particularly that of the cerebellum) that provides a mathematical model of the transformation of sensory space-time coordinates into motor coordinates and vice versa by cerebellar neuronal net ...
is a theory of cerebellar function that provides a mathematical model of the transformation of sensory
space-time In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
coordinates into motor coordinates and vice versa by cerebellar
neuronal networks A neural circuit is a population of neurons interconnected by synapses to carry out a specific function when activated. Neural circuits interconnect to one another to form large scale brain networks. Biological neural networks have inspired the ...
. The theory was developed by Andras Pellionisz and
Rodolfo Llinas Rodolfo is a given name. Notable people with the name include: * Rodolfo (footballer, born 1992), Brazilian footballer Rodolfo José da Silva Bardella * Rodolfo Albano III, Filipino politician * Rodolfo Vera Quizon Sr. (1928-2012), Filipino actor ...
in the 1980s as a geometrization of brain function (especially of the
central nervous system The central nervous system (CNS) is the part of the nervous system consisting primarily of the brain and spinal cord. The CNS is so named because the brain integrates the received information and coordinates and influences the activity of all p ...
) using
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
s.


Software modelling approaches and tools


Neural networks

In this approach the strength and type, excitatory or inhibitory, of synaptic connections are represented by the magnitude and sign of weights, that is, numerical
coefficients In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
w' in front of the inputs x to a particular neuron. The response of the j-th neuron is given by a sum of nonlinear, usually " sigmoidal" functions g of the inputs as: f_=\sum_g\left(w_'x_+b_\right). This response is then fed as input into other neurons and so on. The goal is to optimize the weights of the neurons to output a desired response at the output layer respective to a set given inputs at the input layer. This optimization of the neuron weights is often performed using the backpropagation algorithm and an optimization method such as
gradient descent In mathematics, gradient descent (also often called steepest descent) is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the ...
or Newton's method of optimization. Backpropagation compares the output of the network with the expected output from the training data, then updates the weights of each neuron to minimize the contribution of that individual neuron to the total error of the network.


Genetic algorithms

Genetic algorithms are used to evolve neural (and sometimes body) properties in a model brain-body-environment system so as to exhibit some desired behavioral performance. The evolved agents can then be subjected to a detailed analysis to uncover their principles of operation. Evolutionary approaches are particularly useful for exploring spaces of possible solutions to a given behavioral task because these approaches minimize a priori assumptions about how a given behavior ought to be instantiated. They can also be useful for exploring different ways to complete a computational neuroethology model when only partial neural circuitry is available for a biological system of interest.


NEURON

The
NEURON A neuron, neurone, or nerve cell is an electrically excitable cell that communicates with other cells via specialized connections called synapses. The neuron is the main component of nervous tissue in all animals except sponges and placozoa ...
software, developed at Duke University, is a simulation environment for modeling individual neurons and networks of neurons. The NEURON environment is a self-contained environment allowing interface through its
GUI The GUI ( "UI" by itself is still usually pronounced . or ), graphical user interface, is a form of user interface that allows users to interact with electronic devices through graphical icons and audio indicator such as primary notation, inste ...
or via scripting with hoc or python. The NEURON simulation engine is based on a Hodgkin–Huxley type model using a Borg–Graham formulation. Several examples of models written in NEURON are available from the online database ModelDB.


Embodiment in electronic hardware


Conductance-based silicon neurons

Nervous systems differ from the majority of silicon-based computing devices in that they resemble
analog computer An analog computer or analogue computer is a type of computer that uses the continuous variation aspect of physical phenomena such as electrical, mechanical, or hydraulic quantities (''analog signals'') to model the problem being solved. In ...
s (not
digital data Digital data, in information theory and information systems, is information represented as a string of discrete symbols each of which can take on one of only a finite number of values from some alphabet, such as letters or digits. An exampl ...
processors) and massively
parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster o ...
processors, not sequential processors. To model nervous systems accurately, in real-time, alternative hardware is required. The most realistic circuits to date make use of analog properties of existing
digital electronics Digital electronics is a field of electronics involving the study of digital signals and the engineering of devices that use or produce them. This is in contrast to analog electronics and analog signals. Digital electronic circuits are usu ...
(operated under non-standard conditions) to realize Hodgkin–Huxley-type models ''in silico''.


Retinomorphic chips

Kwabena Boahen, "A Retinomorphic Chip with Parallel Pathways: Encoding INCREASING, ON, DECREASING, and OFF Visual Signals", Analog Integrated Circuits and Signal Processing, 30, 121–135, 2002


See also

*
Cognitive architecture A cognitive architecture refers to both a theory about the structure of the human mind and to a computational instantiation of such a theory used in the fields of artificial intelligence (AI) and computational cognitive science. The formalized mod ...
*
Cognitive map A cognitive map is a type of mental representation which serves an individual to acquire, code, store, recall, and decode information about the relative locations and attributes of phenomena in their everyday or metaphorical spatial environment. T ...
*
Computational neuroscience Computational neuroscience (also known as theoretical neuroscience or mathematical neuroscience) is a branch of neuroscience which employs mathematical models, computer simulations, theoretical analysis and abstractions of the brain to u ...
*
Motion perception Motion perception is the process of inferring the speed and direction of elements in a scene based on visual, vestibular and proprioceptive inputs. Although this process appears straightforward to most observers, it has proven to be a difficult ...
* Neural coding * Neural correlate *
Neural decoding Neural decoding is a neuroscience field concerned with the hypothetical reconstruction of sensory and other stimuli from information that has already been encoded and represented in the brain by networks of neurons. Reconstruction refers to the ...
*
Neuroethology Neuroethology is the evolutionary and comparative approach to the study of animal behavior and its underlying mechanistic control by the nervous system. It is an interdisciplinary science that combines both neuroscience (study of the nervous syste ...
*
Neuroinformatics Neuroinformatics is the field that combines informatics and neuroscience. Neuroinformatics is related with neuroscience data and information processing by artificial neural networks. There are three main directions where neuroinformatics has to be ...
* Quantitative models of the action potential *
Spiking neural network Spiking neural networks (SNNs) are artificial neural networks that more closely mimic natural neural networks. In addition to neuronal and synaptic state, SNNs incorporate the concept of time into their operating model. The idea is that neuron ...
*
Systems neuroscience Systems neuroscience is a subdiscipline of neuroscience and systems biology that studies the structure and function of neural circuits and systems. Systems neuroscience encompasses a number of areas of study concerned with how nerve cells behave ...


References


External links


Neural Dynamics at NSI
– Web page of Patrick D Roberts at the Neurological Sciences Institute
Dickinson Lab
– Web page of the Dickinson group at Caltech which studies flight control in ''Drosophila'' {{DEFAULTSORT:Models Of Neural Computation Ethology Computational neuroscience Neuroethology