Kleene's T predicate

In computational neuroscience, the Wilson–Cowan model describes the dynamics of interactions between populations of very simple excitatory and inhibitory model neurons. It was developed by H.R. Wilson and Jack D. Cowan and extensions of the model have been widely used in modeling neuronal populations. The model is important historically because it uses phase plane methods and numerical solutions to describe the responses of neuronal populations to stimuli. Because the model neurons are simple, only elementary limit cycle behavior, i.e. neural oscillations, and stimulus-dependent evoked responses are predicted. The key findings include the existence of multiple stable states, and hysteresis, in the population response.

Mathematical description

The Wilson–Cowan model considers a homogeneous population of interconnected neurons of excitatory and inhibitory subtypes. The fundamental quantity is the measure of the activity of an excitatory or inhibitory subtype within the population. More precisely, $E(t)$ and $I(t)$ are respectively the proportions of excitatory and inhibitory cells firing at time t. They depend on the proportion of sensitive cells (that are not refractory) and on the proportion of these cells receiving at least threshold excitation.

Sensitive cells

Proportion of cells in refractory period (absolute refractory period $r$) $\int _{t-r}^{t}E(t')dt'$

Proportion of sensitive cells (complement of refractory cells) $1-\int _{t-r}^{t}E(t')dt'$

Excited cells

Subpopulation response function based on $D(\theta )$ the distribution of neuronal thresholds

$S(x)=\int _{0}^{x(t)}D(\theta )d\theta$

Subpopulation response function based on the distribution of afferent synapses per cell (all cells have the same threshold)

$S(x)=\int _{\frac {\theta }{x(t)}}^{\infty }C(w)dw$

Average excitation level

$\int _{-\infty }^{t}\alpha (t-t')[c_{1}E(t')-c_{2}I(t')+P(t')]dt'$

where $\alpha (t)$ is the stimulus decay function, $c_{1}$ and $c_{2}$ are respectively the connectivity coefficient giving the average number of excitatory and inhibitory synapses per cell, P(t) is the external input to the excitatory population.

Excitatory subpopulation expression

$E(t)=[1-\int _{t-r}^{t}E(t')dt']S(x)dt$

Complete Wilson–Cowan model

$E(t+\tau )=[1-\int _{t-r}^{t}E(t')dt']S_{e}\left\{\int _{-\infty }^{t}\alpha (t-t')[c_{1}E(t')-c_{2}I(t')+P(t')]dt'\right\}$

$I(t+\tau )=[1-\int _{t-r}^{t}I(t')dt']S_{i}\left\{\int _{-\infty }^{t}\alpha (t-t')[c_{3}E(t')-c_{4}I(t')+Q(t')]dt'\right\}$

Time Coarse Graining $\tau {\frac {d{\bar {E}}}{dt}}=-{\bar {E}}+(1-r{\bar {E}})S_{e}[kc_{1}{\bar {E}}(t)+kP(t)]$

Isocline Equation $c_{2}I=c_{1}E-S_{e}^{-1}\left({\frac {E}{k_{e}-r_{e}E}}\right)+P$

Sigmoid Function $S(x)={\frac {1}{1+\exp[-a(x-\theta )]}}-{\frac {1}{1+\exp(a\theta )}}$

References

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