Thermodynamic Order Parameters and Statistical-Mechanical Measures for Characterization of the Burst and Spike Synchronizations of Bursting Neurons

By varying the noise intensity, we investigate the population synchronization in an inhibitory population of bursting Hindmarsh-Rose neurons. Unlike spiking neurons, bursting neurons show firing patterns with two timescales: a fast spiking timescale and a slow bursting timescale that modulates the spiking activity. Through separation of the fast and slow timescales, we characterize the burst and spike synchronization transitions by using “thermodynamic” order parameters, and quantitatively measure the degree of the burst and spike synchronizations by employing “statistical-mechanical” measures. Population synchronization may be well visualized in the raster plot of neural spikes which can be obtained in experiments. Instantaneous population firing rate, R(t), which is directly obtained from the raster plot of spikes, is a realistic population quantity showing collective behaviors with both the slow and fast timescales. Through frequency filtering, we separate R(t) into Rb(t) (the instantaneous population burst rate (IPBR) describing the bursting behavior) and Rs(t) (the instantaneous population spike rate (IPSR) describing the intraburst spiking behavior). The time-averaged fluctuations of Rb and Rs play the role of thermodynamic order parameters, Ob and Os, used for characterizing the burst and spike synchronization transitions, respectively. For more direct visualization of bursting behavior, we consider another raster plot of bursting onset or offset times, from which we can directly obtain the IPBR, R (on) b (t) or R (off) b (t), without frequency filtering. Time-averaged fluctuations of R (on) b (t) and R (off) b (t) are also shown to play the role of the order parameters, O b and O (off) b , for the bursting transition. Furthermore, the degree of burst synchronization seen in the raster plot of bursting onset or offset times is well measured in terms of a statisticalmechanical bursting measure Mb, introduced by considering the occupation and the pacing patterns of bursting onset or offset times. Similarly, we also develop Corresponding Author. Email addresses: [email protected] (Sang-Yoon Kim), [email protected] (Woochang Lim) Preprint submitted to Elsevier May 11, 2014 a statistical-mechanical spiking measure Ms, based on Rs, to make practical characterization of the intraburst spike synchronization. It is thus shown that both thermodynamic order parameters and statistical-mechanical measures are effectively used to characterize the burst and spike synchronizations of bursting neurons.