A computational neural model may be constrained to the level of biochemical signalling in individual
neurons 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 potentials in neurons are initiated and propagated in axons via
voltage-gated ion channels. It is a set of
nonlinear ordinary differential equations that were introduced by
Alan Lloyd Hodgkin and
Andrew Huxley in 1952 to explain the results of
voltage clamp experiments on the
squid giant axon. Analytic solutions do not exist, but the
Levenberg–Marquardt algorithm, a modified
Gauss–Newton algorithm, is often used to
fit these equations to voltage-clamp data. The
FitzHugh–Nagumo model is a simplification of the Hodgkin–Huxley model. The
Hindmarsh–Rose model 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 generators.
Solitons The
soliton model is an alternative to the
Hodgkin–Huxley model that claims to explain how
action potentials are initiated and conducted in the form of certain kinds of
solitary sound (or
density) pulses that can be modeled as
solitons along
axons, based on a thermodynamic theory of nerve pulse propagation.
Transfer functions and linear filters This approach, influenced by
control theory and
signal processing, 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, 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 filters turns out to be useful in characterizing neural circuitry. Both
low- and
high-pass filters 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 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, increases the contrast and sharpness in visual response, but leads to the epiphenomenon of
Mach bands. This is often illustrated by the
optical illusion 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 cells. 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_{p}=\left|\left[e_{p}-\sum_{j=1,j\ne p}^{n}k_{pj}\left|r_{j}-r_{pj}^{o}\right|\right]\right|. Here, e_p is the excitation of the target
p-th receptor from sensory transduction r_{pj}^o is the associated threshold of the firing cell, and k_{pj} is the coefficient of inhibitory interaction between the
p-th and the
jth 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 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. Following Fischer and Anderson, the response of the postsynaptic neuron to the signals from the left and right ears is given by y_{R}\left(t\right) - y_{L}\left(t\right) where y_{L}\left(t\right)=\int_{0}^{\tau}u_{L}\left(\sigma\right)w\left(t-\sigma\right)d\sigma y_{R}\left(t\right)=\int_{0}^{\tau}u_{R}\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 ===
Neurophysiological metronomes: neural circuits for pattern generation Mutually
inhibitory processes are a unifying motif of all
central pattern generators. 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.
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, a pair of knob-like organs which measure angular velocity. Integrated computer models of
Drosophila, short on neuronal circuitry but based on the general guidelines given by
control theory 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 is a theory of
cerebellar function that provides a mathematical model of the
transformation of sensory
space-time coordinates into motor coordinates and vice versa by cerebellar
neuronal networks. The theory was developed by Andras Pellionisz and
Rodolfo Llinas in the 1980s as a
geometrization of brain function (especially of the
central nervous system) using
tensors. ==Software modelling approaches and tools==