Abstract
We analyse periodic solutions in a system of four delayed differential equations forced by periodic inputs representing two competing neural populations connected with fast mutual excitation and slow delayed inhibition. The combination of mechanisms generates a rich dynamical structure that we are able to characterize using slow-fast dissection and a binary classification of states. We previously proved the existence conditions of all possible states 1:1 locked to the inputs and applied this analysis to the tracking of the rhythms perceived when listening to alternating sequences of low and high tones. Here we extend this analysis using analytical and computational tools by proving the existence a set of n:1 periodically locked states and their location in parameter space. Firstly we examine cycle skipping states and find that they accumulate in an infinite cascade of period-incrementing bifurcations with increasing periods for decreasing values of the local input strength. Secondly we analyse periodic solutions that alternate between 1:1 locked states that repeat after an integer multiple of the input period (swapping states). We show that such states accumulate in similar bifurcation cascades with decreasing values of the lateral input strength. We report a parameter-dependent scaling constant for the ratio of widths of successive regions in the cascades, which generalises across cycle skipping and swapping states. The periodic states reported here - emergent behaviours in the model - can be linked to known phenomena in auditory perception that are beyond the original scope of the model’s design.
Highlights
Differentiating between sound sources that overlap or are interleaved in time is a fundamental part of auditory perception [1, 2]
A (B) is OFF at time t* 5 A (B) is OFF ∀t ∈. We used this theorem to define the matrix form of each state, which describes the units’ dynamics in one active tone interval R. We extended this representation for two active tone intervals I1A and I1B to describe the dynamics of 2TR-periodic states and use it to determine the existence conditions for these states
We extend the analysis of switching states Wm,k using a continuous version of model (1), described by the following system of six ordinary differential equations (ODEs): τ u_A (t)
Summary
Differentiating between sound sources that overlap or are interleaved in time is a fundamental part of auditory perception (the so-called cocktail party problem) [1, 2]. Slow-fast analysis including singular perturbation theory has been instrumental in revealing the dynamical mechanisms behind spiking and bursting [12, 14] and in explaining complex dynamics in population models of neural competition [10, 15] Extensions of these methods have been applied to systems with delays [16], in non-smooth settings [17] and in networks capable of instrinsically generating patterns of rhythmic behaviours (so-called central pattern generators, CPGs), such as locomotion, breathing, sleep [18]. This combination of modelling assumptions (fixed delays, slow-fast timescale separation, heaviside activation function) allows for all possible model states to be conveniently represented in a binary matricial form with entries specifying the state of the system in time intervals relating to the input timecourse, delay and activation timescales This approach allows for exact parameterdependent existence conditions to be derived, as for all 1:1 locked states in [6] and for more exotic states as reported here.
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