Phase-locking patterns in a resonate and fire neural model with periodic drive
Introduction
As everyone knows, the brain is one of the best examples of a nonlinear complex system, composed of several interacting parts organized in a well structured hierarchical order (Kandel et al., 2000). Nevertheless, in the last years many important results have been obtained on relevant processes regulating its currently working activity. However, we are far from a complete picture elucidating the general functional architecture of the brain over the different spatial and temporal time scales. Indeed, coherently with the above remarks, a clear example where our knowledge should be improved concerns the information processing occurring just at the single neuron level. In particular, the understanding of the basic mechanisms, driving the responses of a neuronal cell to its incoming stimulation patterns, is of fundamental importance for advancing our knowledge of the processes regulating the coding of sensory information also in small neural networks (Kheradpezhouh et al., 2017; Singer, 2018). Pacemaker neurons operate in a characteristic frequency band and they are basic units of neural circuits controlling rhythmic motion (swimming, locomotion, heart beat, peristaltic motions, etc.) (Grillner, 2006). Neurons exhibiting pacemaker behaviour are found also in the visual cortex. For instance chattering cells generate gamma oscillations when depolarized by current injection (Gray and McCormick, 1996). Neurons behave as relaxation oscillators because after the spike there is a time window (refractory period) in which the basic physiological concentrations of ionic species should be recovered. In neural circuits a regular oscillatory activity of neurons can be also generated by specific negative feedback networks (or recurrent inhibition). The basic mechanism, underlying the generation of such synchronous oscillatory electric activity, arises from the reciprocal synaptic interaction between excitatory and inhibitory neurons (Buzsáki and Wang, 2012). Indeed, interneurons are capable of synchronizing their firing activity and that of the principal cells by means of powerful inhibition (Crandall and Connors, 2016). Moreover, synchronization phenomena between the oscillatory electric activities of different neural populations have been observed experimentally(Gilbert and Wiesel, 1989; Lowel and Singer, 1992; Bosking et al., 1997; Pecka et al., 2014; Gray et al., 1989; Schillen and König, 1994; Singer, 1999; Uhlhaas et al., 2009). Now, for the aims of the present work, it is useful to point out what follows. Let be, for instance, C1 and C2 two columns of the visual cortex (but can also be others specific neural circuits) and assume that they are generating oscillatory electrical activities (for instance generated by some of the mechanisms discussed before) at principal (or dominant) frequencies ω1 and ω2, respectively. Now, two closely spaced columns responding to similar orientation, are coupled and therefore column 1 receive an oscillatory inputs with frequency ω2 arising in column 2 and vice versa. Thus, simplifying the matter, each column can be thought as a nonlinear oscillator receiving periodic inputs. This is one example of neurophysiological system that justifies the research work that will be carried out in this paper. Another important way in which synchronization phenomena and phase-locking occur is when techniques involving electromagnetic stimulations are employed (Kheradpezhouh et al., 2017). More precisely, the study of the single neuron response to periodic electrical inputs is relevant in the context of transcranial magnetic-acoustical stimulations, were the neural tissue is stimulated locally with periodic electrical signals (Thut et al., 2012). Therefore, in such case, it is important to assess which kind of responses is generated by the single neuron. From the experimental point of view such studies are not simple to carry out, although some progress has been made. Thus, to this aim, an important methodological strategy consists in the study of phase-locking patterns in simple neuronal mathematical models subject to special inputs like the periodic ones (Rabinovich et al., 2006; Yuan et al., 2017). From a theoretical point of view the investigation of such phase-locking phenomena can be employed to recover information on the ongoing mechanisms underlying the observed behavior (Pikovsky et al., in press). Moreover, to reduce the intrinsic complexity of these problems, a useful strategy is that of employing simplified, but still realistic, neuron models, such as the integrate-and-fire class (Keener et al., 1981; Lánský, 1997; Brette, 2004; Laudanski et al., 2009; Engelbrecht et al., 2013). On the other hand, to the best of our knowledge, the studies of phase-locking phenomena in more realistic model like the resonate and fire ones, is just at the beginnings. This class of model is even a very simplified version of more complex models, but still remaining analytically tractable. These models (describing the subthreshold dynamics) are characterized by two coupled time dependent variables, one describing the time evolution of the membrane potential and the other playing the role of recovering, and they are linear models (Hill, 1936; Izhikevich, 2001). In keeping with the above discussion, in this paper we study the responses of a resonate and fire neural model, proposed in Di Garbo et al. (2001), to periodic inputs. This model was obtained by linearization of the well-known FitzHugh-Nagumo (FHN) neural model in a neighbour of its fixed point (Izhikevich, 2007). This model will be denoted as LFHN and in Di Garbo et al. (2001) it was studied in the case in which such fixed point was unstable: i.e. after a supercritical Hopf bifurcation occurred. Here we study the more realistic situation in which the corresponding fixed point is stable. An interesting topic, worth investigating for resonate and fire models, is that of studying and analyzing the properties of the firing map. Recently a similar work was done for a general class of integrate and fire model in Marzantowicz and Signerska (2015), Signerska-Rynkowska (2015). In the present contribution, using some tools employed to study orientation preserving map on the circle, we will study the phase-locking phenomena occurring in the perturbed LFHN model either in the absence/presence of noise.
Section snippets
Description of FHN and LFHN models
In the FHN model the state of a neuron is defined by two variables: one (V) representing the membrane potential of the neuron and the other (W) describing the repolarization phase after the action potential generation. As usual the variable V is the faster and the corresponding mathematical model is defined by the following equations:where a = 0.5,ϵ = 0.005 and b is a bifurcation parameter (Longtin, 1993). The stationary states s = (Vs, Ws), or fixed points of
Subthreshold dynamics of LFHN
In the following we are interested to describe the subthreshold dynamics of the LFHN. In other words we search for the explicit time dependent expression of (x(t), y(t)) with initial conditions: (x(t0), y(t0)) = (xr, yr). The eigenvalues associated to the LFHN equations 3,d, are the solutions of the equation . By setting D2 = (R + ϵ)2 − 4ϵ it is easy to show that the eigenvalues are real when D2 ≥ 0. When this inequality does not hold the corresponding eigenvalues are complex
Solutions of the LFHN for constant stimulation currents
Before discussing the conditions leading to LFHN firing, it is useful to take briefly in consideration the orbits structure of equation 3,4 in absence of resetting. To this aim in Fig. 3 are plotted the phase portraits of such dynamical system for the case B(t) = Is. In the left panel are reported the orbits for several initial conditions (identified by filled circles) in the case Is = 0. It can be seen that the motion along the orbits is in clockwise direction and this implies that a typical
Firing conditions of LFHN for a time dependent perturbation
Let us now consider a more complex situation in which the perturbation is not constant. In what follows it will be shown that, assuming realistic conditions on the property of the function B(t), it is possible to find general condition to get the firing for the LFHN model. Let us assume that B(t) = Is + P(t), where P(t) is not necessarily periodic. Moreover, let assume that P(t) ∈ C(R) and it is bounded |P(t)| ≤ L. Using these assumptions it is possible to prove the following proposition: Theorem 5.1 Let us
Conclusions
As discussed at the beginning, an important strategy for improving our knowledge on the mechanisms underlying the signal transduction of sensory information from single neurons is to investigate the synchronization properties of these cells both experimentally and theoretically (Singer, 2018). In the present work this target was pursued by studying phase-locking phenomena in a periodically forced resonate and fire model. In particular, the LFHN neural model, obtained by linearizing the FHN
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