voltage_fluctuation

class brainpy.measure.voltage_fluctuation(potentials, numpy=True, method='loop')

Calculate neuronal synchronization via voltage variance.

The method comes from [1] [2] [3].

First, average over the membrane potential \(V\)

\[V(t) = \frac{1}{N} \sum_{i=1}^{N} V_i(t)\]

The variance of the time fluctuations of \(V(t)\) is

\[\sigma_V^2 = \left\langle \left[ V(t) \right]^2 \right\rangle_t - \left[ \left\langle V(t) \right\rangle_t \right]^2\]

where \(\left\langle \ldots \right\rangle_t = (1 / T_m) \int_0^{T_m} dt \, \ldots\) denotes time-averaging over a large time, \(\tau_m\). After normalization of \(\sigma_V\) to the average over the population of the single cell membrane potentials

\[\sigma_{V_i}^2 = \left\langle\left[ V_i(t) \right]^2 \right\rangle_t - \left[ \left\langle V_i(t) \right\rangle_t \right]^2\]

one defines a synchrony measure, \(\chi (N)\), for the activity of a system of \(N\) neurons by:

\[\chi^2 \left( N \right) = \frac{\sigma_V^2}{ \frac{1}{N} \sum_{i=1}^N \sigma_{V_i}^2}\]

[1]

Golomb, D. and Rinzel J. (1993) Dynamics of globally coupled inhibitory neurons with heterogeneity. Phys. Rev. E 48:4810-4814.

[2]

Golomb D. and Rinzel J. (1994) Clustering in globally coupled inhibitory neurons. Physica D 72:259-282.

[3]

David Golomb (2007) Neuronal synchrony measures. Scholarpedia, 2(1):1347.

Parameters:

• potentials – The membrane potential matrix of the neuron group.

• numpy – Whether we use numpy array as the functional output. If False, this function can be JIT compiled.

• method –

  The method to calculate all pairs of cross correlation.

  Supports two kinds of methods: loop and vmap.

  vmap method will consume much more memory.

  New in version 2.2.3.4.

Returns:

The synchronization index.

Return type:

sync_index