Variation Statistics for CBGTC spike times

class analyseur.cbgtc.stats.variation.Variations[source]

Bases: object

Computes variation or dispersion in the data

Methods

Argument

computeCV()

isi_set: Dictionary returned; see InterSpikeInterval

computeCV2()

isi_set: Dictionary returned; see InterSpikeInterval

computeLV()

isi_set: Dictionary returned; see InterSpikeInterval

grandCV()

isi_set: Dictionary returned; see InterSpikeInterval

grandCV2()

isi_set: Dictionary returned; see InterSpikeInterval

grandLV()

isi_set: Dictionary returned; see InterSpikeInterval

Use Cases

1. Pre-requisites

1.1. Import Modules

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

1.2. Load file and get spike times

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

1.3. Compute the Inter-Spike Intervals

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

2. Cases

2.1. Compute for variations

CV = Variations.computeCV(I)
CV2 = Variations.computeCV2(I)
LV = Variations.computeLV(I)

This returns the respective values for

  • \(\overrightarrow{CV} = \left[cv^{(i)}\right]_{\forall{i \in [1, n_{nuc}]}}\)

  • \(\overrightarrow{CV_2} = \left[cv_2^{(i)}\right]_{\forall{i \in [1, n_{nuc}]}}\)

  • \(\overrightarrow{LV} = \left[lv^{(i)}\right]_{\forall{i \in [1, n_{nuc}]}}\)

2.2. Compute Global Values

grand_CV = Variations.grandCV(I)
grand_CV2 = Variations.grandCV2(I)
grand_LV = Variations.grandLV(I)

This returns the respective values for

  • \(\overline{CV} = \mu\left(\overrightarrow{CV}\right)\)

  • \(\overline{CV_2} = \mu\left(\overrightarrow{CV_2}\right)\)

  • \(\overline{LV} = \mu\left(\overrightarrow{LV}\right)\)

classmethod computeCV(isi_set=None)[source]
\[ \begin{align}\begin{aligned}\overrightarrow{cv}^{(i)} = \begin{cases} \text{NaN} & \text{if } n_{ISI}^{(i)} = 0 \\ \frac{\sqrt{\frac{1}{n_{ISI}^{(i)}} \sum_{k=1}^{n_{ISI}^{(i)}} \left(isi_k^{(i)} - \mu_i\right)^2 }}{\text{max}\left(\frac{1}{n_{ISI}^{(i)}} \sum_{k=1}^{n_{ISI}^{(i)}} isi_k^{(i)}, 10^{-8} \right)} & \text{otherwise} \end{cases}\end{aligned}\end{align} \]

Returns the coefficient of variation for all individual neurons.

Parameters:

isi_set – Dictionary returned using analyseur.cbgtc.stats.isi.InterSpikeInterval.compute()

Returns:

dictionary of individual neurons whose values are their respective coefficient of variation value

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 coefficient of variation of the i-th neuron is

\[cv^{(i)} = \frac{\sigma\left(\overrightarrow{ISI}^{(i)}\right)}{\mu\left(\overrightarrow{ISI}^{(i)}\right)}\]

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

We therefore get

Definitions

Interpretation

\(cv^{(i)}\)

coefficient of variation of i-th neuron

\(\overrightarrow{CV} = \left[cv^{(i)}\right]_{\forall{i \in [1, n_{nuc}]}}\)

array of coefficient of variation of all neurons
classmethod computeCV2(isi_set=None)[source]
\[ \begin{align}\begin{aligned}\overrightarrow{cv}_2^{(i)} = \begin{cases} \text{NaN} & \text{if } n_{ISI}^{(i)} \le 1 \\ \frac{1}{\left(n_{ISI}^{(i)} - 1\right)} \sum_{k=2}^{n_{ISI}^{(i)}} \left[2 \cdot \frac{\left| isi_k^{(i)} - isi_{k-1}^{(i)} \right|}{\text{max}\left(isi_k^{(i)} - isi_{k-1}^{(i)}, 10^{-8} \right)}\right] & \text{otherwise} \end{cases}\end{aligned}\end{align} \]

Returns the local coefficient of variation for all individual neurons.

Parameters:

isi_set – Dictionary returned using analyseur.cbgtc.stats.isi.InterSpikeInterval.compute()

Returns:

dictionary of individual neurons whose values are their respective local coefficient of variation value

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 coefficient of variation of the i-th neuron is

\[cv_2^{(i)} = \mu\left(2\frac{\left[\left|isi^{(i)}_k - isi^{(i)}_{k-1}\right|\right]_{\forall{k \in [1, n_{spk}^{(i)})}}}{\left[isi^{(i)}_k + isi^{(i)}_{k-1}\right]_{\forall{k \in [1, n_{spk}^{(i)})}}}\right)\]

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

We therefore get

Definitions

Interpretation

\(cv_2^{(i)}\)

local coefficient of variation of the i-th neuron

\(\overrightarrow{CV_2} = \left[cv_2^{(i)}\right]_{\forall{i \in [1, n_{nuc}]}}\)

array of local coefficient of variation of all neurons
classmethod computeLV(isi_set=None)[source]
\[ \begin{align}\begin{aligned}\overrightarrow{lv}^{(i)} = \begin{cases} \text{NaN} & \text{if } n_{ISI}^{(i)} \le 1 \\ \frac{1}{\left(n_{ISI}^{(i)} - 1\right)} \sum_{k=2}^{n_{ISI}^{(i)}} \left[3 \cdot \frac{\left(isi_k^{(i)} - isi_{k-1}^{(i)}\right)^2}{\text{max}\left(\left(isi_k^{(i)} - isi_{k-1}^{(i)}\right)^2, 10^{-8} \right)}\right] & \text{otherwise} \end{cases}\end{aligned}\end{align} \]

Returns the local variation for all individual neurons.

Parameters:

isi_set – Dictionary returned using analyseur.cbgtc.stats.isi.InterSpikeInterval.compute()

Returns:

dictionary of individual neurons whose values are their respective local variation value

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 local variation of the i-th neuron is

\[lv^{(i)} = \mu\left(3\frac{\left[\left(isi^{(i)}_k - isi^{(i)}_{k-1}\right)^2\right]_{\forall{k \in [1, n_{spk}^{(i)})}}}{\left[\left(isi^{(i)}_k + isi^{(i)}_{k-1}\right)^2\right]_{\forall{k \in [1, n_{spk}^{(i)})}}}\right)\]

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

We therefore get

Definitions

Interpretation

\(lv^{(i)}\)

local variation of the i-th neuron

\(\overrightarrow{LV} = \left[lv^{(i)}\right]_{\forall{i \in [1, n_{nuc}]}}\)

array of local coefficient of variation of all neurons
classmethod grandCV(isi_set=None)[source]

Returns the grand coefficient of variation which is the mean of coefficient of variation of all the neurons

Parameters:

isi_set – Dictionary returned using analyseur.cbgtc.stats.isi.InterSpikeInterval.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

coefficient of variation, \(\overline{cv^{(i)}}\)

coefficient of variation of i-th neuron

\(\overrightarrow{CV} = \left[cv^{(i)}\right]_{\forall{i \in [1, n_{nuc}]}}\)

array of coefficient of variation of all neurons

grand coefficient of variation, \(\overline{CV} = \mu\left(\overrightarrow{CV}\right)\)

grand or global coefficient of variation

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

NOTE: The array \(\overrightarrow{CV}\) is obtained by calling computeCV()

classmethod grandCV2(isi_set=None)[source]

Returns the grand local coefficient of variation which is the mean of local coefficient of variation of all the neurons

Parameters:

isi_set – Dictionary returned using analyseur.cbgtc.stats.isi.InterSpikeInterval.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

local coefficient of variation, \(\overline{cv^{(i)}}\)

local coefficient of variation of i-th neuron

\(\overrightarrow{CV_2} = \left[cv_2^{(i)}\right]_{\forall{i \in [1, n_{nuc}]}}\)

array of local coefficient of variation of all neurons

grand local coefficient of variation, \(\overline{CV_2} = \mu\left(\overrightarrow{CV_2}\right)\)

grand or global local coefficient of variation

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

NOTE: The array \(\overrightarrow{CV_2}\) is obtained by calling computeCV2()

classmethod grandLV(isi_set=None)[source]

Returns the grand local variation which is the mean of local variation of all the neurons

Parameters:

isi_set – Dictionary returned using analyseur.cbgtc.stats.isi.InterSpikeInterval.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

local variation, \(\overline{cv^{(i)}}\)

local variation of i-th neuron

\(\overrightarrow{LV} = \left[lv^{(i)}\right]_{\forall{i \in [1, n_{nuc}]}}\)

array of local variation of all neurons

grand local variation, \(\overline{LV} = \mu\left(\overrightarrow{LV}\right)\)

grand or global local variation

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

NOTE: The array \(\overrightarrow{LV}\) is obtained by calling computeLV()