Active spike transmission in the neuron model with a winding threshold manifold V.B. Kazantsev a,n , A.S. Tchakoutio Nguetcho b,c , S. Jacquir b , S. Binczak b , J.M. Bilbault b a Institute of Applied Physics of RAS, 46 Uljanov Street, 603950 Nizhny Novgorod, Russia b Laboratoire LE2I UMR CNRS 5158, Universite´ de Bourgogne, BP 47870, 21078 Dijon Cedex, France c Departement des Sciences Physiques, Ecole Normale Superieure, de Maroua, Universite´ de Maroua, BP 46, Maroua, Cameroun article info Article history: Received 1 December 2010 Received in revised form 20 December 2011 Accepted 24 December 2011 Communicated by W.L. Dunin-Barkowski Available online 3 January 2012 Keywords: Excitability Spike transmission Active response Spike encoding Threshold manifold Nonlinear dynamics abstract We analyze spiking responses of excitable neuron model with a winding threshold manifold on a pulse stimulation. The model is stimulated with external pulse stimuli and can generate nonlinear integrate- and-fire and resonant responses typical for excitable neuronal cells (all-or-none). In addition we show that for certain parameter range there is a possibility to trigger a spiking sequence with a finite number of spikes (a spiking message) in the response on a short stimulus pulse. So active transformation of N incoming pulses to M (with M 4N) outgoing spikes is possible. At the level of single neuron computations such property can provide an active ‘‘spike source’’ compensating ‘‘spike dissipation’’ due to the integrate-and-fire N to 1 response. We delineate the dynamical mechanism for the N to M transformation based on the winding threshold manifold in the neighborhood of big saddle loop bifurcation. Based on the theoretical predictions, a nonlinear electronic circuit is designed implement- ing the active transmission in physical conditions. & 2012 Elsevier B.V. All rights reserved. 1. Introduction The principles of inter-neuron communication, representation of sensory information and its processing in neuronal networks still remain among the key problems in understanding brain dynamics and cognitive functions [1–7]. According to many experimental works and theoretical reviews in modern neu- roscience, neuronal networks in the brain form spontaneous and stimulus-induced patterns of spiking activity (spiking patterns). Information and functions are believed to be encoded in the dynamical characteristics of the patterns and their evolution. In time such patterns represent spiking sequences with variable inter-spike intervals (ISI). Sensory signals incoming to the brain act as external stimuli allowing the current configuration to adapt to current environmental conditions. Such processes underly many brain functions, for example associative memory formation, learning, perception, motor control, etc. [3]. It is believed that neurons communicate using excitation pulses (spikes) [1,5,6,8–11]. The information can be encoded in the characteristics of spiking sequences including variable inter- spike interval, modulation of spike frequency or spiking phase relative to an oscillatory drive (rhythm) [12,13]. At the level of a single neuron, the problem of signal processing leads to the analysis of the cell response on different input pulse stimuli. The input pulse can be interpreted either as a single, strong synaptic input or as a packet of synchronously arriving inputs, such as occurs in synfire chains or polychrony. The responses are defined by the dynamics of cell membrane excitability (threshold characteristics, fast and slow time scales, etc.), by the strength and timing of the input. Basic properties of the neuron response include threshold firing by integrating the incoming perturba- tions (integrate-and-fire) and resonance filtering when the response spikes appear only at favorable frequencies and phases of the incoming signal. Many realistic neuron models have been proposed and explored in detail in a number of previous studies [5,10,11,14–17]. The gallery of the simplest models and their response features compared with electrophysiological recordings from different brain cells has been presented in [5]. When stimulated by periodic pulse signals, the excitable models can generate the response oscillations at a certain level of forcing synchronized (or quasi-periodic) with the input at different phase locking modes ending up with 1:1 response for strong forcing [16]. Many simplified phenomenological neuron models compris- ing these features have been proposed and used in the large network modeling [18–20]. Due to the excitability at the level of single neuron the number of output spikes, M, is typically less than the number of integrated inputs, N, M ¼ 1 rN. Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/neucom Neurocomputing 0925-2312/$ - see front matter & 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.neucom.2011.12.014 n Corresponding author. Tel.: þ7 8314164905; fax: þ7 8314363792. E-mail addresses: vkazan@neuron.appl.sci-nnov.ru (V.B. Kazantsev), sjacquir@u-bourgogne.fr (S. Jacquir), stbinc@u-bourgogne.fr (S. Binczak), bilbault@u-bourgogne.fr (J.M. Bilbault). Neurocomputing 83 (2012) 205–211