Biological neuron networks have reverberating loops; inferior olive (IO).

Periodic large-amplitude oscillations can happen even with each individual neurons firing at a significantly smaller rate or of irregular spike trains. Nicely explained in Fig. 8.11

Strong oscillations with irregular spike trains is related to short-term memory and timing tasks.

Binary neurons: potential of the ith neuron at time $t_{n+1}$ is determined by the states of other neurons at time $t_n$ $u_i(t_{n+1})=w_{ij}S_j(t_n)$. The state of neuron $S_i(t_n)$ is determined by the potential at time $t_{n}$, $S_i(t_n)=\Theta(u_i(t_n)-\theta)$, where $\theta$ is threshold.

Approximate SRM to McClulloch-Pitts neurons with “digitized” states.

For sparsely connect we can approximate the time evolution using independent events and find the probability.

Fig. 8.12; The interactions are shown on the top panels. We start from a value of $a_n$, the iteration gives us the result of $a_{n+1}$. Then the next step depends on the value of $a_{n+1}$ so we project it onto the dashed line $a_{n}=a_{n+1}$. Then we use the new $a_n$ value to find the new $a_{n+1}$.

Excitations and Inhibitions

Random network with balanced excitations and inhibitions can generate broad interval distributions.

Reverberating projections usually have both excitation and inhibitions.

McClulloch-Pitts model with both excitations and inhibitions.

Microscopic Dynamics

The simplified model (SRM->McClulloch-Pitts) doesn’t catch all the features, with inhibitory neurons in presence. The limit circle can grow substantially larger as size of the network increases.