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 simplication 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.^[8][9]

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.^[10] Assuming these interactions to be linear, they proposed the following relationship for the steady-state response rate ${\displaystyle r_{p}}$ of the given p-th photoreceptor in terms of the steady-state response rates ${\displaystyle r_{j}}$ of the j surrounding receptors:

${\displaystyle r_{p}=\left|\left[e_{p}-\sum _{j=1,j\neq p}^{n}k_{pj}\left|r_{j}-r_{pj}^{o}\right|\right]\right|}$.

Here,

${\displaystyle e_{p}}$ is the excitation of the target p-th receptor from sensory transduction

${\displaystyle r_{pj}^{o}}$ is the associated threshold of the firing cell, and

${\displaystyle k_{pj}}$ 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,^[11] 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,^[12] the response of the postsynaptic neuron to the signals from the left and right ears is given by

${\displaystyle y_{R}\left(t\right)-y_{L}\left(t\right)}$

where

${\displaystyle y_{L}\left(t\right)=\int _{0}^{\tau }u_{L}\left(\sigma \right)w\left(t-\sigma \right)d\sigma }$

${\displaystyle y_{R}\left(t\right)=\int _{0}^{\tau }u_{R}\left(\sigma \right)w\left(t-\sigma \right)d\sigma }$

and

${\displaystyle w\left(t-\sigma \right)}$ 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.^[13]

Cross-correlation for motion detection: Hassenstein–Reichardt model

A motion detector needs to satisfy three general requirements: pair-inputs, asymmetry and nonlinearity.^[14] 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.^[15]

The master equation for response is

${\displaystyle R=A_{1}(t-\tau )B_{2}(t)-A_{2}(t-\tau )B_{1}(t)}$

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.^[16]

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.^[17]

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.^[18] 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.^[19] 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.^[20][21]

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.^[22][23]