Iter-Spiking Interval Statistics for CBGTC spike times

class analyseur.cbgtc.stats.isi.InterSpikeInterval[source]

Bases: object

Computes interspike intervals for the given spike times

Methods

Argument

compute()

inst_rates()

  • isi_set: Dictionary returned; see compute()

pool_avg_inst_rates()

  • inst_rates_set: Dictionary returned; see inst_rates()

  • tbins_set: 2nd tuple (Dictionary) returned; see compute()

  • binsz: [OPTIONAL] 0.01 (default)

mean_freqs()

  • isi_set: Dictionary returned; see compute()

grand_mean_freq()

  • isi_set: Dictionary returned; see compute()

Use Cases

1. Pre-requisites

1.1. Import Modules

from analyseur.cbgtc.loader import LoadSpikeTimes
from analyseur.cbgtc.stats.isi import InterSpikeInterval

1.2. Load file and get spike times

loadST = LoadSpikeTimes("spikes_GPi.csv")
spiketimes_superset = loadST.get_spiketimes_superset()

2. Cases

2.1. Compute Inter-Spike Intervals (for all neurons)

[I, all_t] = InterSpikeInterval.compute(spiketimes_superset)

This returns the value for \(I = \left\{\overrightarrow{ISI}^{(i)} \mid \forall{i \in [1, n_{nuc}]} \right\}\); see formula compute().

2.2. Compute Instantaneuous Rates (for all neurons)

J = InterSpikeInterval.inst_rates(I)

This returns the value for \(R = \left\{\vec{R}^{(i)} \mid \forall{i \in [1, n_{nuc}]} \right\}\); see formula inst_rates()

2.3. Compute Average Instantaneuous Rates (for all neurons)

E = InterSpikeInterval.pool_avg_inst_rates(J, all_t)

This returns the value for \(\vec{\Xi} = [\xi_t]_{\forall{t}}\); see formula avg_pool_inst_rates()

2.4. Compute Mean Frequencies (for all neurons)

F = InterSpikeInterval.mean_freqs(I)

This returns the value for \(\vec{F} = \left[\overline{f^{(i)}}\right]_{\forall{i \in [1, n_{nuc}]}}\); see formula mean_freqs()

2.5. Compute Global Mean Frequency

grand_f = InterSpikeInterval.grand_mean_freq(I)

This returns the value for \(\overline{f}\); see formula grand_mean_freq()

classmethod compute(spiketimes_set=None)[source]
\[ \begin{align}\begin{aligned}\overrightarrow{ISI}^{(i)} = \begin{cases} \varnothing & n_{spk}^{(i)} < 2 \\ \left[t_{k+1}^{(i)} - t_k^{(i)}\right] & \text{otherwise} \end{cases}\end{aligned}\end{align} \]

Returns the interspike interval for all individual neurons.

Parameters:

spiketimes_set – Dictionary returned using get_spiketimes_superset()

or using get_spiketimes_subset()

Returns:

2-tuple

  • dictionary of individual neurons whose values are their respective interspike interval

  • dictionary of individual neurons whose values are their respective times for corresponding interspike interval

Formula

Definitions

Interpretation

total neurons, \(n_{nuc}\)

total number of neurons in the Nucleus

neuron index, \(i\)

i-th neuron in the pool of \(n_{nuc}\) neurons

total spikes, \(n_{spk}^{(i)}\)

total number of spikes (spike times) by i-th neuron

interspike interval, \(isi_{k}^{(i)}\)

k-th absolute interval between successive spike times

\(\overrightarrow{ISI}^{(i)} = \left[isi_k^{(i)}\right]_{\forall{k \in [1, n_{spk}^{(i)})}}\)

array of all interspike intervals of i-th neuron

\(I = \left\{\overrightarrow{ISI}^{(i)} \mid \forall{i \in [1, n_{nuc}]} \right\}\)

set of array interspike intervals of all neurons
classmethod grand_mean_freq(isi_set=None)[source]

Returns the grand mean frequency which is the mean of mean frequencies of all the neurons

Parameters:

isi_set – Dictionary returned using compute()

Returns:

a number

Formula

Definitions

Interpretation

total neurons, \(n_{Nuc}\)

total number of neurons in the Nucleus

neuron index, \(i\)

i-th neuron in the pool of \(n_{Nuc}\) neurons

mean frequency, \(\overline{f^{(i)}}\)

mean spiking frequency of i-th neuron

\(\vec{F} = \left[\overline{f^{(i)}}\right]_{\forall{i \in [1, n_{nuc}]}}\)

array of mean frequencies of all (\(n_{Nuc}\)) neurons

grand mean frequency, \(\overline{f} = \mu\left(\vec{F}\right)\)

grand or global mean spiking frequency

where, \(\mu(\cdot)\) is the arithmetic mean function over the given dimension.

NOTE: The array \(\vec{F}\) is obtained by calling mean_freqs()

classmethod inst_rates(isi_set=None)[source]

Returns the instantaneous rates for all individual neurons.

Parameters:

isi_set – Dictionary returned using compute()

Returns:

dictionary of individual neurons whose values are their respective instantaneous rates

Formula

Definitions

Interpretation

total neurons, \(n_{nuc}\)

total number of neurons in the Nucleus

neuron index, \(i\)

i-th neuron in the pool of \(n_{Nuc}\) neurons

total spikes, \(n_{spk}^{(i)}\)

total number of spikes (spike times) by i-th neuron

interspike interval, \(isi_{k}^{(i)}\)

k-th absolute interval between successive spike times

\(\overrightarrow{ISI}^{(i)} = \left[isi_k^{(i)}\right]_{\forall{k \in [1, n_{spk}^{(i)})}}\)

array of all interspike intervals of i-th neuron

\(I = \left\{\overrightarrow{ISI}^{(i)} \mid \forall{i \in [1, n_{nuc}]} \right\}\)

set of array interspike intervals of all neurons

Then, the instantaneuous rate of i-th neuron is

\[ \begin{align}\begin{aligned}\vec{R}^{(i)} &= \frac{1}{\overrightarrow{ISI}^{(i)}} \\ &= \left[\frac{1}{isi_k^{(i)}}\right]_{\forall{k \in [1, n_{spk}^{(i)})}}\end{aligned}\end{align} \]

We therefore get

Definitions

Interpretation

\(\vec{R}^{(i)}\)

array of instantaneous rates of i-th neuron

\(R = \left\{\vec{R}^{(i)} \mid \forall{i \in [1, n_{nuc}]} \right\}\)

set of array of instaneous rates of all (\(n_{Nuc}\)) neurons
classmethod mean_freqs(isi_set=None)[source]

Returns the mean frequencies for all individual neurons.

Parameters:

isi_set – Dictionary returned using compute()

Returns:

dictionary of individual neurons whose values are their respective mean frequencies

Formula

Definitions

Interpretation

total neurons, \(n_{nuc}\)

total number of neurons in the Nucleus

neuron index, \(i\)

i-th neuron in the pool of \(n_{Nuc}\) neurons

total spikes, \(n_{spk}^{(i)}\)

total number of spikes (spike times) by i-th neuron

interspike interval, \(isi_{k}^{(i)}\)

k-th absolute interval between successive spike times

\(\overrightarrow{ISI}^{(i)} = \left[isi_k^{(i)}\right]_{\forall{k \in [1, n_{spk}^{(i)})}}\)

array of all interspike intervals of i-th neuron

\(I = \left\{\overrightarrow{ISI}^{(i)} \mid \forall{i \in [1, n_{nuc}]} \right\}\)

set of array interspike intervals of all neurons

Then, the mean spiking frequency of i-th neuron is

\[\overline{f^{(i)}} = \frac{1}{(n_{spk}^{(i)} - 1)} \sum_{j=1}^{(n_{spk}^{(i)} - 1)}\frac{1}{isi_{j}^{(i)}}\]

We therefore get

Definitions

Interpretation

\(\overline{f^{(i)}}\)

mean spiking frequency of i-th neuron

\(\vec{F} = \left[\overline{f^{(i)}}\right]_{\forall{i \in [1, n_{nuc}]}}\)

array of mean frequencies of all (\(n_{Nuc}\)) neurons
classmethod pool_avg_inst_rates(inst_rates_set=None, tbins_set=None, binsz=None)[source]

Returns the pooled average instantaneous rates for all individual neurons.

Parameters:

  • tbins_set – Dictionary returned using compute()

  • inst_rates_set – Dictionary returned using inst_rates()

  • binsz – integer or float; 0.01 [default]

Returns:

3-tuple

  • list of average instantaneous rates

  • array of centers for all the time bins (use this as time axis for plotting)

  • list of number of data point per bin (can be useful for colorbar)

Formula

Definitions

Interpretation

total neurons, \(n_{nuc}\)

total number of neurons in the Nucleus

neuron index, \(i\)

i-th neuron in the pool of \(n_{Nuc}\) neurons

\(\vec{R}^{(i)}\)

array of instantaneous rates of i-th neuron

\(R = \left\{\vec{R}^{(i)} \mid \forall{i \in [1, n_{nuc}]} \right\}\)

set of array of instaneous rates of all (\(n_{Nuc}\)) neurons

\(\vec{J}^{(i)}\)

array of time points where instantaneous rates of i-th neuron occur

\(J = \left\{\vec{J}^{(i)} \mid \forall{i \in [1, n_{nuc}]} \right\}\)

set of array of time points of all (\(n_{Nuc}\)) neurons

For using bin-based conditional average let

Definitions

Interpretation

\(\vec{P} = vec(J)\)

array containing all the time points from all the neurons

\(\vec{\Xi} = vec(R)\)

array containing all the instantaneous rates from all the neurons

total bins, \(n_{bins} = \mid vec(J) \mid\)

total number of time points from all neurons

bin size, \(w\)

fixed bin width for each time bin

bin center, \(c_{\forall{t} \in [0, n_{bins} - 1]}\)

center of t-th time bin

bin interval, \(b_t = \left[c_t - \frac{w}{2}, c_t + \frac{w}{2}\right)\)

interval of t-th time bin

Then, the average instantaneuous rate for t-th bin is

\[ \begin{align}\begin{aligned}\xi_t &= \mathbb{E}\left[\Xi_p \mid p \in b_t\right] \\ &= \frac{\sum_{p \in P}(\Xi_p \cdot 1_{\{p \in b_t\}})}{\sum_{p \in P} 1_{\{p \in b_t\}}}\end{aligned}\end{align} \]

where

  • \(\mathbb{E}\) is the expectation function,

  • \(1_{\{p \in b_t\}}\) is the indicator function; 1 if condition is true otherwise 0,

  • \(\sum_{p \in P} 1_{\{p \in b_t\}}\) is the number of time points that fall in the t-th bin

  • numerator is the sum of instantaneous rates that fall in the t-th bin

We therefore get

Definitions

Interpretation

\(\xi_t\)

average instantaneous rate for t-th bin

\(\vec{\Xi} = [\xi_t]_{\forall{t}}\)

array of average instantaneous rates for all bins
classmethod true_avg_inst_rates(inst_rates_set=None)[source]

Returns the true average instantaneous rates for all individual neurons.

Parameters:

  • tbins_set – Dictionary returned using compute()

  • inst_rates_set – Dictionary returned using inst_rates()

  • binsz – integer or float; 0.01 [default]

Returns:

3-tuple

  • list of average instantaneous rates

  • array of centers for all the time bins (use this as time axis for plotting)

  • list of number of data point per bin (can be useful for colorbar)

Formula

Definitions

Interpretation

total neurons, \(n_{nuc}\)

total number of neurons in the Nucleus

neuron index, \(i\)

i-th neuron in the pool of \(n_{Nuc}\) neurons

\(\vec{R}^{(i)}\)

array of instantaneous rates of i-th neuron

\(R = \left\{\vec{R}^{(i)} \mid \forall{i \in [1, n_{nuc}]} \right\}\)

set of array of instaneous rates of all (\(n_{Nuc}\)) neurons

\(\vec{J}^{(i)}\)

array of time points where instantaneous rates of i-th neuron occur

\(J = \left\{\vec{J}^{(i)} \mid \forall{i \in [1, n_{nuc}]} \right\}\)

set of array of time points of all (\(n_{Nuc}\)) neurons

For using bin-based conditional average let

Definitions

Interpretation

\(\vec{P} = vec(J)\)

array containing all the time points from all the neurons

\(\vec{\Xi} = vec(R)\)

array containing all the instantaneous rates from all the neurons

total bins, \(n_{bins} = \mid vec(J) \mid\)

total number of time points from all neurons

bin size, \(w\)

fixed bin width for each time bin

bin center, \(c_{\forall{t} \in [0, n_{bins} - 1]}\)

center of t-th time bin

bin interval, \(b_t = \left[c_t - \frac{w}{2}, c_t + \frac{w}{2}\right)\)

interval of t-th time bin

Then, the average instantaneuous rate for t-th bin is

\[ \begin{align}\begin{aligned}\xi_t &= \mathbb{E}\left[\Xi_p \mid p \in b_t\right] \\ &= \frac{\sum_{p \in P}(\Xi_p \cdot 1_{\{p \in b_t\}})}{\sum_{p \in P} 1_{\{p \in b_t\}}}\end{aligned}\end{align} \]

where

  • \(\mathbb{E}\) is the expectation function,

  • \(1_{\{p \in b_t\}}\) is the indicator function; 1 if condition is true otherwise 0,

  • \(\sum_{p \in P} 1_{\{p \in b_t\}}\) is the number of time points that fall in the t-th bin

  • numerator is the sum of instantaneous rates that fall in the t-th bin

We therefore get

Definitions

Interpretation

\(\xi_t\)

average instantaneous rate for t-th bin

\(\vec{\Xi} = [\xi_t]_{\forall{t}}\)

array of average instantaneous rates for all bins