28. Oscillations in Reverberating Loops

Outlines

1. Biological neuron networks have reverberating loops; inferior olive (IO).
2. 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
3. Strong oscillations with irregular spike trains is related to short-term memory and timing tasks.
4. 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.
5. Approximate SRM to McClulloch-Pitts neurons with “digitized” states.
6. For sparsely connect we can approximate the time evolution using independent events and find the probability.
7. 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

0. Random network with balanced excitations and inhibitions can generate broad interval distributions.
1. Reverberating projections usually have both excitation and inhibitions.
2. McClulloch-Pitts model with both excitations and inhibitions.

Microscopic Dynamics

1. 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.
2. Information will drain away with noise. Fig. 8.15