Rate Statistics of CBGTC spike times¶

class analyseur.cbgtc.stats.rate.Rate[source]¶

Bases: object

Use Cases¶

1. Pre-requisites¶

1.1. Import Modules¶

from analyseur.cbgtc.loader import LoadSpikeTimes
from analyseur.cbgtc.stats.rate import Rate

1.2. Load file and get spike times¶

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

2. Cases¶

2.1. Compute spike count observables¶

count_matrix, rate_matrix, time_bins = Rate.get_count_rate_matrix(
                                          spiketimes_set=spiketimes_superset)

2.2. Compute spike count observables within a desired window for a select neurons¶

count_matrix, rate_matrix, time_bins = Rate.get_count_rate_matrix(
                                          spiketimes_set=spiketimes_superset,
                                          window=(3,8),
                                          binsz=0.5,
                                          neurons=[2, 3, 6, 7])

2.3. Compute mean firing rate from all given neurons across all neurons¶

mu_rate_vec, time_bins = Rate.mean_rate(
                            spiketimes_set=spiketimes_superset,
                            neurons="all",
                            across="neurons")

2.4. Compute mean firing rate from all given neurons across all times¶

mu_rate_vec, time_bins = Rate.mean_rate(
                            spiketimes_set=spiketimes_superset,
                            neurons="all",
                            across="times")

classmethod get_count_rate_matrix(spiketimes_set=None, window=None, binsz=None, neurons='all')[source]¶

Returns the spike count and firing rate matrices for all neurons.

Parameters:: spiketimes_set – Dictionary returned using get_spiketimes_superset()

or using get_spiketimes_subset()

Parameters:

binsz – integer or float; 0.01 [default]
window – Tuple in the form (start_time, end_time); (0, 10) [default]
neurons –
“all” [default] or scalar or range(a, b) or list of neuron ids like [2, 3, 6, 7]
- ”all” means subset = superset
- N (a scalar) means subset of first N neurons in the superset
- range(a, b) or [2, 3, 6, 7] means subset of selected neurons

Returns:

spike count matrix, firing rate matrix and time bins

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
spike time, \(s^{(i)}\)	an arbitrary spike time of i-th neuron
bin size, \(\beta\)	width of each time bin as discrete interval
function, \(I(x) = \{1 \text{ if } x \text{ is true else } 0\}\)	indicator function which equals 1 if condition inside is true and 0 otherwise
spike count, \(c^{(i)}(t) = \sum_{k} I(t \cdot \beta \le s^{(i)}_k < (t+1) \cdot \beta)\)	number of spikes of the i-th neuron at time \(t\)
spike count matrix, \(C = \left[c(a,b) = c^{(a)}(b)\right]_{\forall{a \in [1, n_{Nuc}], b \in [t_0, t_T]}}\)	spike counts of all (\(n_{Nuc}\)) neurons for all times
firing rate, \(f^{(i)}(t) = \frac{c^{(i)}(t)}{\beta}\)	firing rate of the i-th neuron at time \(t\) with bin size \(\beta\)
firing rate matrix, \(F = \left[f(a,b) = f^{(a)}(b)\right]_{\forall{a \in [1, n_{Nuc}], b \in [t_0, t_T]}}\)	rates of all (\(n_{Nuc}\)) neurons for all times

Spike Count Matrix

\[ \begin{align}\begin{aligned}C = \overset{\begin{matrix}t_0 & \quad\quad & t_1 & & & &\ldots & & & t_T\end{matrix}} {\begin{bmatrix} c^{(1)}(t_0) & c^{(1)}(t_1) & \ldots & c^{(1)}(t_T) \\ c^{(2)}(t_0) & c^{(2)}(t_1) & \ldots & c^{(2)}(t_T) \\ \vdots & \vdots & \ldots & \vdots \\ c^{(i)}(t_0) & c^{(i)}(t_1) & \ldots & c^{(i)}(t_T) \\ \vdots & \vdots & \ldots & \vdots \\ c^{(n_{Nuc})}(t_0) & c^{(n_{Nuc})}(t_1) & \ldots & c^{(n_{Nuc})}(t_T) \end{bmatrix} }\end{aligned}\end{align} \]

Firing Rate Matrix

\[ \begin{align}\begin{aligned}F = \overset{\begin{matrix}t_0 & \quad\quad & t_1 & & & &\ldots & & & t_T\end{matrix}} {\begin{bmatrix} f^{(1)}(t_0) & f^{(1)}(t_1) & \ldots & f^{(1)}(t_T) \\ f^{(2)}(t_0) & f^{(2)}(t_1) & \ldots & f^{(2)}(t_T) \\ \vdots & \vdots & \ldots & \vdots \\ f^{(i)}(t_0) & f^{(i)}(t_1) & \ldots & f^{(i)}(t_T) \\ \vdots & \vdots & \ldots & \vdots \\ f^{(n_{Nuc})}(t_0) & f^{(n_{Nuc})}(t_1) & \ldots & f^{(n_{Nuc})}(t_T) \end{bmatrix} }\end{aligned}\end{align} \]

classmethod mean_rate(spiketimes_set=None, window=None, binsz=None, neurons='all', across=None)[source]¶

Returns the mean firing rate from all neurons.

Parameters:: spiketimes_set – Dictionary returned using get_spiketimes_superset()

or using get_spiketimes_subset()

Parameters:

binsz – integer or float; 0.01 [default]
window – Tuple in the form (start_time, end_time); (0, 10) [default]
neurons –
“all” or scalar or range(a, b) or list of neuron ids like [2, 3, 6, 7]
- ”all” means subset = superset
- N (a scalar) means subset of first N neurons in the superset
- range(a, b) or [2, 3, 6, 7] means subset of selected neurons
across – “neurons” or “times”

Returns:

mean rate and time bins

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
frequency, \(f^{(i)}(t)\)	frequency of the i-th neuron at time \(t\)
frequency matrix, \(F = \left[f(a,b) = f^{(a)}(b)\right]_{\forall{a \in [1, n_{Nuc}], b \in [t_0, t_T]}}\)	frequencies of all (\(n_{Nuc}\)) neurons for all times

Formula: Mean Rate across = “neurons”

\[ \begin{align}\begin{aligned}F = \overset{\begin{matrix}t_0 & \quad\quad & t_1 & & & &\ldots & & & t_T\end{matrix}} {\underset{ \begin{matrix} \quad\quad\downarrow & \quad\quad\quad & \downarrow & \quad &\ldots & & & \downarrow \\ \quad\mu_{t_0} & \quad\quad\quad & \mu_{t_1} & \quad &\ldots & & & \mu_{t_T} & \quad \quad \end{matrix}} {\begin{bmatrix} f^{(1)}(t_0) & f^{(1)}(t_1) & \ldots & f^{(1)}(t_T) \\ f^{(2)}(t_0) & f^{(2)}(t_1) & \ldots & f^{(2)}(t_T) \\ \vdots & \vdots & \ldots & \vdots \\ f^{(i)}(t_0) & f^{(i)}(t_1) & \ldots & f^{(i)}(t_T) \\ \vdots & \vdots & \ldots & \vdots \\ f^{(n_{Nuc})}(t_0) & f^{(n_{Nuc})}(t_1) & \ldots & f^{(n_{Nuc})}(t_T) \end{bmatrix} }}\end{aligned}\end{align} \]

Formula: Mean Rate across = “times”