Firing Pattern Analysis

SULAB: Characterization of Neuronal Firing Patterns

[version 1.1 Dec. 26th, 2016] This page is created by Yasuhiro Mochizuki in collaboration with Shigeru Shinomoto.

Web Application for Analysis of Firing Patterns

1. Copy-and-paste your spike data.

Arrange spike times in ascending order in units of second ( ).

2.

• Average firing characteristics computed for the entire spike train:

Mean Firing Rate:	log λ	=
	λ	=	Hz
Mean Firing Regularity:	log κ	=
Serial Correlation of Interspike Intervals (ISIs):	ρ	=
Local Variation of ISIs:	Lv	=

• Time dependence of firing characteristics (computed for every 100 ISIs.)

3.

CAUTION: If your spike train is bursting (i.e., log κ < 1 and Lv > 1), this web application may return NAN for the value of log κ.
This is due to the numerical instability in a special function provided by Jmat library, but can be avoided by using Matlab code provided below.

Matlab code

Code for analysis of firing patterns: getfp.mat

Function "getfp" returns firing characteristics (firing rate, firing regularity, ISI correlation, and Lv) of a given spike train.

Example usage:
>> ISI = exprnd(exp(-2.0), 1, 2000); spkt = horzcat(0.0, cumsum(ISI));

>> fp=getfp(spkt);

This example simulates a Poisson spike train (log λ = 2, log κ = 0, and ρ = 0) containing 2,001 spikes (the first line), and computes its firing characteristics (the second line). Here, value of n is set to be 20 by default, but can be adjusted by adding the second argument to the input.

For more details about the usage of "getfp", see the comments in Matlab file.

Method

•Firing rate, firing regularity, and ISI correlation
A given spike train is divided to m segments of n = 20 ISIs, and each measurement are made for each segment. The results from all the segments were averaged for each spike train. The firing rate and firing regularity were quantified by fitting the gamma distribution (Kuffler et al., 1957; Stein, 1965; Reich et al., 1998; Brown et al., 2002; Shimokawa and Shinomoto, 2009):

$g_{\lambda,\kappa}(I)=(\lambda\kappa)^{\kappa}I^{\kappa-1}e^{-\lambda \kappa I}/\Gamma(x),$

where κ and λ are the shape and scale factors, respectively, and Γ(x) is the gamma function.

i. Firing rate. The maximum likelihood estimation of the scale factor λ of the gamma distribution gives the firing rate as

$\lambda=\frac{n}{\sum_{i=1}^{n}I_{i}},$

where I_i is the duration of the ith ISI. We indicate the firing activity in terms of log firing rate averaged over m segments, as

$<\log\ \lambda>=\frac{1}{m}\sum_{j=1}^{m}\log\ \lambda_{j},$

where λ_j represents the firing rate in the jth segment. We take the logarithm of the firing rate λ as well as the shape parameter κ, because the information distance between different gamma distributions scales approximately with log λ and log κ (Miura et al., 2006).

ii. Firing regularity. The shape factor κ of the gamma distribution represents the firing regularity. This is estimated by maximizing the likelihood, which is achieved by solving the equation,

$\log \kappa - \psi(\kappa)=\log \Big(\frac{1}{n}\sum_{i=1}^{n}I_{i}}\Big)-\frac{1}{n}\sum_{i=1}^{n}\log\ I_{i},$

where ψ(x)is the digamma function (Cox and Lewis, 1966). We represent the regularity of a given spike train by averaging the logarithm of the shape factor over m segments,

$<\log\ \kappa>=\frac{1}{m}\sum_{j=1}^{m}\log\ \kappa_{j}.$

Firing regularity log κ is expected to take a value of 0 (i.e., κ = 1) for a Poisson spike train, and becomes positive or negative if firings are more regular (κ > 1) or bursting (κ < 1), respectively.

iii. ISI correlation. We estimated the correlation of consecutive ISIs using Spearman’s rank-order correlation of order one (Kuffler et al., 1957; Farkhooi et al., 2009). For each segment of a spike train or a sequence of ISIs {I₁, I₂, … , I_n}, we indicated their rank order as {r₁, r₂, … , r_n}, where r_i denotes the rank of the ith ISI I_i in ascending order. When there were sets of identical ISIs, they were each assigned the average rank. Spearman’s rank-order correlation of order one is defined as

$\rho=\frac{n}{n-1}\ \frac{\sum_{k=1}^{n-1}(r_k-\bar{r})(r_{k+1}-\bar{r})}{\sum_{k=1}^{n}(r_k-\bar{r})},$

where r is the average of ranks (=(n+1)/2). By averaging the rank order over m segments, we represented the degree of renewality of a given spike train of 2,000 ISIs,

$<\rho>=\frac{1}{m}\sum_{j=1}^{m}\rho_{j}.$

See Mochizuki et al., 2016 for more details about firing rate, firing regularity, and ISI correlation.

• Local variation
The index of local variation of ISIs, Lv (Shinomoto et al., 2003), is given as

$Lv=\frac{3}{n-1}\sum_{i=1}^{n-1}\Big(\frac{I_i-I_{i+1}}{I_{i}+I_{i+1}}\Big)^2.$

See Shinomoto et al., 2003 for more details about Lv.

Reference

Materials on this page are based on the following papers:

Similarity in neuronal firing regimes across mammalian species.
Y. Mochizuki, T. Onaga, H. Shimazaki, T Shimokawa, Y. Tsubo, R. Kimura, A. Saiki, Y. Sakai, Y. Isomura, S. Fujisawa, K. Shibata, D. Hirai, T. Furutat . Kaneko, S. Takahashi, T. Nakazono, S. Ishino, Y. Sakurai, T. Kitsukawa, J. W. Lee, H. Lee, M. W. Jung, C. Babul, P. E. Maldonado, F. I. Arce-McShane, K. Takahashi, C. F. Ross, B. J. Sessle, N. G. Hatsopoulos, T. Brochier, A. Riehle, P. Chorley, S. Grün, H. Nishijo, S. Ichihara-Takeda, S. Funahashi, K. Shima, H. Mushiake, Y. Yamane, H. Tamura, I. Fujita, N. Inaba, K. Kawano, S. Kurkin, K. Fukushima, K. Kurata, M. Taira, K. Tsutsui, T. Ogawa, H. Komatsu, K. Koida, K. Toyama, B. J. Richmond, and S. Shinomoto.
Journal of Neuroscience (2016) 36:5736-5747. Supplementary materials.

Differences in spiking patterns among cortical neurons.
S. Shinomoto, K. Shima, and J. Tanji.
Neural Computation (2003) 15:2823-2842.

A part of the web application uses:
Jmat library (Copyright (c) 2011-2016, Lode Vandevenne. All rights reserved.)
JS code provide by Neath I..

If you have any questions, comments, or suggestions for improving the programs, please contact Shigeru Shinomoto who conducts these studies.