27. Synchronized Oscillations and Locking

Terms:

1. Locking: stable state when all neurons are firing.

Finding Locking in Noise-free populations

Verify that locking is a stable solution of the population activity equation.

In an ideal model, the consequence of locking is a fully synchronized firing, which leads to a step-like population activity.

So we assume the population activity has the form of step-like functions, c.f. Eq. (8.11). The visualization of the expression is shown in Fig. 8.4.

At time $t=0$, we have a rectangle. If the rectangle becomes wider, the synchronization will eventually disappear. So for stable synchronization, we would have them becoming more synchronized and obtaining smaller width.

To understand Eq. (8.11), we can consider each mode.

For $k=0$ mode, we have

$$ \frac{1}{2\delta_0} \Theta(t-\delta_0)\Theta(\delta_0 - t). $$

There are three regions:

1. $t<\delta_0$: it’s 0;
2. $t>\delta_0$: it’s 0;
3. $-\delta_0<t<\delta_0$: it’s 1.

Stable locking of all neurons: related to the instantaneous slope of input potential $h$ at the moment of firing. Eq. (8.12)

$$ h’(T)>0 , \Rightarrow , A(T) > A(0). $$

Locking theorem: kind of makes sense.

Examples of Locking

Perfect synchronization in noiseless SRM0.

Consider the case where all neurons are fired at $t=0$. Fig. 8.5.

To derive Eq. (8.16) from Eq. (8.15), we Taylor expand both sides in terms of small deviations $\delta_0$ and $\delta_1$,

$$ \begin{align} &\theta - \eta(T+\delta_1-\delta_0) = J_0 \epsilon(T+\delta_1) \
\Rightarrow & \theta - [ \eta(T) + \eta’(T)(\delta_1-\delta_0) ] = J_0 [\epsilon(T) + \epsilon’(T)\delta_1] \
\Rightarrow & - \eta’(T)(\delta_1-\delta_0) = \epsilon’(T)\delta_1 \
\Rightarrow & \epsilon’(T) = -\frac{ \eta’(T)(\delta_1-\delta_0)}{\delta_1} \end{align} $$

For stability, we need $\delta_1<\delta_0$, i.e., the delay for the next firing is smaller than the delay for the first firing.

SRM0 with inhibitory coupling

Fig. 8.6

Locking with Noise

Noise will smear out the spikes. So a system with synchronization have to be with small noise.

Otherwise the condition should be no different from noiseless case.

Cluster states

1. Fig. 8.9
2. Conditions for stable locking for each clustering subgroup: Eq. (8.30)
3. Asynchronous firing: smear out the peaks and distribute spikes evenly in time.
4. Neurons that fire with a delay will be pulled back to the subgroup that just synchronously fired. But neurons that fired earlier than the synchronous firing will not be pulled back to this subgroup but rather the subgroup that fired before.
5. Due to noise, some neurons will drift off one subgroup and be pulled into another subgroup.