Date Published: March 21, 2017
Publisher: Public Library of Science
Author(s): Sorinel A. Oprisan, Dave I. Austin, Maurice J. Chacron.
We derived analytically and checked numerically a set of novel conditions for the existence and the stability of phase-locked modes in a biologically relevant master-slave neural network with a dynamic feedback loop. Since neural oscillators even in the three-neuron network investigated here receive multiple inputs per cycle, we generalized the concept of phase resetting to accommodate multiple inputs per cycle. We proved that the phase resetting produced by two or more stimuli per cycle can be recursively computed from the traditional, single stimulus, phase resetting. We applied the newly derived generalized phase resetting definition to predicting the relative phase and the stability of a phase-locked mode that was experimentally observed in this type of master-slave network with a dynamic loop network.
Oscillatory neural activity is ubiquitous and covers a wide spatial and temporal scale from single neural cells to whole brain regions and from milliseconds to days. Neural oscillations are believed to be relevant for a wide range of brain activities from sensory information processing to consciousness . It is believed that the phase of low frequency theta oscillations (4-8 Hz) drives the pyramidal cells and is used for information processing in the hippocampus [2–4]. Visual stimuli binding is believed to be related to the phase resetting of the fast frequency gamma band (30-70 Hz) . Positive phase correlations between the theta rhythm and the amplitude of gamma oscillations were found during visual stimuli processing and learning [1, 6, 7] and during fear-related information processing [8, 9]. Theta rhythm resetting also drives cognitive processes . Theoretical studies suggested that phase resetting could explain cross-frequency phase-locking of gamma rhythm within a theta cycle , which is the hallmark of successful memory retrieval [12, 13]. The phase of neural oscillations is also used to bridge a much wider frequency range from slow theta rhythms of large neural networks, such as those in the hippocampus, up to the individual fast spiking neurons used for speech decoding . It was found that speech resets background (rest) oscillatory activity in specific frequency domains corresponding to the sampling rates optimal for phonemic and syllabic sampling [14, 15]. Phase resetting is also critical in the functioning of suprachiasmatic nucleus that produces a stable circadian oscillation by light-induced resetting of endogeneous rhythm [16, 17]. It was also shown that single sensory stimulus [18, 19] and periodic train of inputs [20, 21] induce phase resetting in electroencephalograms, which manifest as event-related evoked potentials.
The phase response curve (PRC) method has been extensively used for predicting phase-locked modes in neural networks [51–54]. It assumes that the only effect of a stimulus is to reset the phase of the ongoing oscillation of a neuron. Traditionally, the PRC tabulates the transient change in the firing frequency of a neural oscillator in response to one external stimulus per cycle of oscillation. The term PRC has been used almost exclusively in regard to a single stimulus per cycle of neural oscillators. Recently, we suggested a generalization of the PRC that allowed us to account for the overall resetting when two or more inputs are delivered during the same cycle . As a result, we expanded the PRC theory from the prediction of the traditional one-to-one phase-locked modes to arbitrary phase-locked firing patterns. Here we present the first quantitative application of such generalized PRC approach to a realistic neural network with a dynamic feedback loop.
In their seminal work on giant squid axon, Hodgkin and Huxley [60–64] experimentally identified three classes, or types, of axonal excitability: class I, where the repetitive firing is controlled by the intensity of an external stimulus; class II, where the firing frequency is almost independent on stimulus intensity; and class III, where there are no endogenous bursters regardless of stimulus intensity or duration.
In order to use the PRC method (see section 2) for predicting the relative phases of neurons in a phase-locked firing pattern, we assumed a fixed firing order of the three neurons with the goal of determining if such a pattern exists and if it is stable. Based on the neural network model proposed for delayed and anticipated synchronization by Matias et al [40–42], we identified the following definitions for the firing period of each neuron (see Fig 3):
where t2r is the recovery time of neuron #2 after its last input, t2sa and t2sb are the corresponding stimulus times for the first and, respectively, the second input to neuron #2, and the index of the cycle is marked with the square brackets […]. The subscript index refers to the neural oscillator index according to Fig 3. From Eq (4) we eliminated t2r[n − 1] = P1i − t2sb[n] and substituted it into the other two equations, which led to:
Let us assume that there is a steady state solution (t2sa*,t2sb*) for the recursive Eq (6) that mimics the activity of the neural network shown in Fig 3, i.e. the following limits exist limn∞t2sa[n]=t2sa* and limn∞t2sa[n]=t2sa*. By substituting the steady state, i.e. phase-locked mode, solution (t2sa*,t2sb*) into Eq (6) one obtains:
where we used the fact that t3s*=t2sb*-t2sa*+t2r* and that t2r[n − 1] = P1i − t2sb[n], which led to t3s*=t2sb*-t2sa*+P1i-t2sb*=P1i-t2sa*.
The possible phase-locked modes given by Eq (6) may not all be stable and, therefore, they may not be all experimentally observable. To determine the stability of the steady solutions (t2sa*,t2sb*), we assume small perturbations:
where the nth cycle perturbation δt2s[n]<