Derivation of a neural field model from a network of theta neurons

Neural field models are used to study macroscopic spatiotemporal patterns in the cortex. Their derivation from networks of model neurons normally involves a number of assumptions, which may not be correct. Here we present an exact derivation of a neural field model from an infinite network of theta neurons, the canonical form of a type I neuron. We demonstrate the existence of a “bump” solution in both a discrete network of neurons and in the corresponding neural field model.

Figure 1

A simulation of (1, 2, 3, 4) with coupling function $K\left(x\right)=0.1+0.3cosx$. (a) $sin\theta$, color-coded. (b) Snapshot of the network at $t=100$. (c) Average firing frequency over $0\le t\le 100$. Parameters: $L=2\pi ,k=2,N=600,n=2,{\eta }_0=-0.4,\Delta =0.02$.

Figure 2

Steady state of (19). Compare with Fig. 1. (a) Magnitude of $z$. (b) Argument of $z$. (c) Frequency, as given by (22). Parameters: $L=2\pi ,k=2,n=2,{\eta }_0=-0.4,\Delta =0.02$.

Figure 3

(a) Maximum firing frequency in the network as a function of ${\eta }_0$. Solid: stable. Dashed: unstable. (b) Frequency profiles at the two points marked in (a) (solid blue curve, blue circle; dashed red curve, red square). Parameters: $L=2\pi ,k=2,n=2,\Delta =0.02$.