International Journal of Non-Linear Mechanics 44 (2009) 432--440 Contents lists available at ScienceDirect International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm A new method for characterizing patterns of neural spike trains and its application Ying Du a,b , Qishao Lu a, ∗ , Shimin Wang a,b , Marian Wiercigroch b a Division of General Mechanics, School of Aeronautical Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, PR China b Center for Applied Dynamics Research, School of Engineering, University of Aberdeen, Kings College, Aberdeen AB24 3UE, Scotland, UK ARTICLE INFO ABSTRACT Article history: Received 22 July 2008 Received in revised form 23 December 2008 Accepted 27 January 2009 Keywords: Neural spike trains Counting process Inter-spike-interval Patterns A method for characterizing and identifying firing patterns of neural spike trains is presented. Based on the time evolution of a neural spike train, the counting process is constructed as a time-dependent stair-like function. Three characteristic variables defined at sequential moments, including two formal derivatives and the integration of the counting process, are introduced to reflect the temporal patterns of a spike train. The reconstruction of a spike train with these variables verify the validity of this method. And a model of cold receptor is used as an example to investigate the temporal patterns under different temperature conditions. The most important contribution of our method is that it not only can reflect the features of spike train patterns clearly by using their geometrical properties, but also it can reflect the trait of time, especially the change of bursting of spike train. So it is a useful complementarity to conventional method of averaging. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction It was well known that neural information processing relies on the transmission of a series of stereotyped events called action po- tentials or spikes. Temporal recording of firing events of a neuron provides an inter-spike-interval (ISI) series. It is expected that the processed neural information can be encoded in the structure of ISI series, that is, the neural firing activities can be represented by the patterns of neural spike trains. The basic biophysics underlying the generation of action potentials (spikes) is somewhat well established, but the encoding mechanism is still unclear. At present, there are many existing methods to explore how neu- ral information respond to different stimuli. Many researchers mea- sured the statistical significance of temporal structures in spike trains in order to determine how much information about stimuli contained in neural responses by means of the information theory [1,2]. For example, to investigate the encoding meaning of spike timing and the temporal rhythm structures (that is, patterns) of spikes, the se- ries expansion approximation method [3], the information distortion method [4] and other methods have also been used to quantify the information encoded in spike trains [5–10]. These works indicated that the recognition of temporal patterns of a spike train is essential for extracting information from neuronal responses; however, the features of conveyed information have not been well understood yet. ∗ Corresponding author. E-mail address: qishaolu@hotmail.com (Q. Lu). 0020-7462/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2009.01.007 Therefore, it is important to develop some other methods to recog- nize the temporal patterns of spike trains, that is, to understand the information represented by the neuron. Several characteristic vari- ables were proposed to quantify the structure of spike trains [11], but their geometric meaning and the use in the reconstruction of spike trains are still not clear. Taking into account of the above situation, an approach to identify the temporal structures of spikes in neural systems is developed in this paper. The key feature of this method is to introduce the counting process, which exhibits the dynamic characteristic variables of spike trains. Several characteristics of spike train are deduced from the counting process at sequential moments and used to describe the temporal pattern of a spike train. The comparison with the usual tuning curve method is also developed. This paper is organized as follows. The counting process with its formal derivatives and integration are introduced to characterize the temporal pattern of a spike train in Section 2. An example for the response of a cold receptor and the comparison with previous methods are presented in Section 3, and the conclusion is given in Section 4. 2. Methodology and counting process 2.1. Counting process and its first formal derivative A recorded neural spike train can be characterized by a bounded variation function q(t) called counting process as follows: q(t) = i, t i  t<t i+1 (i = 1, 2, . . .), (1)