Patterns, Memory and Periodicity in Two-Neuron Delayed Recurrent Inhibitory Loops

Abstract

Abstract. We study the coexistence of multiple periodic solutions for an analogue of the integrateand-fire neuron model of two-neuron recurrent inhibitory loops with delayed feedback, which incorporates the firing process and absolute refractory period. Upon receiving an excitatory signal from the excitatory neuron, the inhibitory neuron emits a spike with a pattern-related delay, in addition to the synaptic delay. We present a theoretical framework to view the inhibitory signal from the inhibitory neuron as a self-feedback of the excitatory neuron with this additional delay. Our analysis shows that the inhibitory feedbacks with firing and the absolute refractory period can generate four basic types of oscillations, and the complicated interaction among these basic oscillations leads to a large class of periodic patterns and the occurrence of multistability in the recurrent inhibitory loop. We also introduce the average time of convergence to a periodic pattern to determine which periodic patterns have the potential to be used for neural information transmission and cognition processing in the nervous system.

Emergence and combinatorial accumulation of jittering regimes in spiking oscillators with delayed feedback

“…The study on this basic and very common "motif" [43] is important for understanding the behavior of larger delay-coupled networks [44]. For instance, a loop consisting of one excitatory and one inhibitory neuron with delayed connection shows similar behavior to a neuron with delayed self-feedback [45][46][47], and also the behavior of rings of several neurons are, in some cases, related to the behavior of a single neuron with delayed feedback [48][49][50]. In fact, a larger neuronal feedback delay might firstly arise due to a chained propagation of action potentials along a ring of neurons.…”

“…Although we approach the problem of an oscillator with pulsed feedback from a general perspective, not bound to a specific area of application, neurons with delayed feedback are a prototypical example for such systems. There exist some studies on this subject [41,[45][46][47]53], all of which have a common baseline: delay leads arXiv:1512.03567v1 [nlin.CD] 11 Dec 2015 to immense multistability. At first glance this result is not too surprising since multistability generically arises in delay differential equations due to a well-known mechanism called "periodic solution reappearance" [54].…”

Quantitative Neuroscience: From Chalk Board to Bedside

“…In other words, it is not only necessary to develop analytical expressions that describe neural synchronization regardless of the number of neurons [5], it is also necessary to understand the effects of noise of the dynamics of synchronizing neural populations [2]. Similarly the widespread occurrence of time-delayed feedback in neural pathways raises questions as to the role of time delays in information processing [10] and whether new effects arise from the interplay between noise …”

“…Brain-computer interfaces now make it possible to translate thought into action [6,7,8,15], replace lost limbs with robotic ones [9], prevent epileptic seizures [11,16] and even to alleviate the symptoms of neuro-degenerative diseases, such as Parkinson's [14]. Is it possible to do even better?The first theme of this issue explores two lines of mathematical research that have been particularly fruitful: 1) neural synchronization [19,21], the fundamental process by which spatially distributed neural centers bind a sensory stimulus and coordinate their activities to respond to it, and 2) multistability [2,10,12], the concept that treatments may be possible by applying jolts of electricity to the right location in the brain at the right time [11,16]. Attention is drawn to the effects of time delays and random perturbations ("noise") on neural control and information processing.…”

“…The first theme of this issue explores two lines of mathematical research that have been particularly fruitful: 1) neural synchronization [19,21], the fundamental process by which spatially distributed neural centers bind a sensory stimulus and coordinate their activities to respond to it, and 2) multistability [2,10,12], the concept that treatments may be possible by applying jolts of electricity to the right location in the brain at the right time [11,16]. Attention is drawn to the effects of time delays and random perturbations ("noise") on neural control and information processing.…”