Effect of Alpha-Type external input on annihilation of self-sustained activity in a two population neural field model

In the present work, we investigate the annihilation of persistent localized activity states (bumps) in a Wilson-Cowan type two-population neural field model in response to $\alpha$ -type spatio-temporal external input. These activity states serves as working memory in the prefrontal cortex. The impact of different parameters involved in the external input on annihilation of these persistent activity states is investigated in detail. The $\alpha$ -type temporal function in the external input is closer to natural phenomenon as observed in Roth et. al . ( Nature Neuroscience , vol. 19 (2016), 229–307). Two types of eraser mechanism are used in this work to annihilate the spatially symmetric solutions. Initially, if there is an activity in the network, inhibitory external input with no excitatory part and over excitation with no inhibition in the external input can kill the activity. Our results show that the annihilation of persistent activity states using $\alpha$ -type temporal function in the external input is more roubust and more efficient as compare to triangular one as used by Yousaf et al. ( Neural networks. , vol. 46 (2013), pp. 75–90). It is also found that the relative inhibition time constant plays a crucial role in annihilation of the activity. Runge-Kutta fourth order method has been employed for numerical simulations of this work.publishedVersio

Many Attractors, Long Chaotic Transients, and Failure in Small-World Networks of Excitable Neurons

We study the dynamical states that emerge in a small-world network of recurrently coupled excitable neurons through both numerical and analytical methods. These dynamics depend in large part on the fraction of long-range connections or `short-cuts' and the delay in the neuronal interactions. Persistent activity arises for a small fraction of `short-cuts', while a transition to failure occurs at a critical value of the `short-cut' density. The persistent activity consists of multi-stable periodic attractors, the number of which is at least on the order of the number of neurons in the network. For long enough delays, network activity at high `short-cut' densities is shown to exhibit exceedingly long chaotic transients whose failure-times averaged over many network configurations follow a stretched exponential. We show how this functional form arises in the ensemble-averaged activity if each network realization has a characteristic failure-time which is exponentially distributed.Comment: 14 pages 23 figure

Dynamical principles in neuroscience

Dynamical modeling of neural systems and brain functions has a history of success over the last half century. This includes, for example, the explanation and prediction of some features of neural rhythmic behaviors. Many interesting dynamical models of learning and memory based on physiological experiments have been suggested over the last two decades. Dynamical models even of consciousness now exist. Usually these models and results are based on traditional approaches and paradigms of nonlinear dynamics including dynamical chaos. Neural systems are, however, an unusual subject for nonlinear dynamics for several reasons: (i) Even the simplest neural network, with only a few neurons and synaptic connections, has an enormous number of variables and control parameters. These make neural systems adaptive and flexible, and are critical to their biological function. (ii) In contrast to traditional physical systems described by well-known basic principles, first principles governing the dynamics of neural systems are unknown. (iii) Many different neural systems exhibit similar dynamics despite having different architectures and different levels of complexity. (iv) The network architecture and connection strengths are usually not known in detail and therefore the dynamical analysis must, in some sense, be probabilistic. (v) Since nervous systems are able to organize behavior based on sensory inputs, the dynamical modeling of these systems has to explain the transformation of temporal information into combinatorial or combinatorial-temporal codes, and vice versa, for memory and recognition. In this review these problems are discussed in the context of addressing the stimulating questions: What can neuroscience learn from nonlinear dynamics, and what can nonlinear dynamics learn from neuroscience?This work was supported by NSF Grant No. NSF/EIA-0130708, and Grant No. PHY 0414174; NIH Grant No. 1 R01 NS50945 and Grant No. NS40110; MEC BFI2003-07276, and Fundación BBVA

The spatiotemporal dynamics of human focal seizures

Spontaneous human focal seizures can present with a plethora of behavioral manifestations that vary according to the affected cortical regions; however, several key features have been consistently observed. During my doctoral studies, I applied both theoretical and experimental methods to study mechanisms underpinning these consistently seen dynamics. I first analyzed human intracranial EEG recordings, describing statistical methods for measuring their electrophysiological signatures. I next proposed several neurophysiological hypotheses that could explain seizure dynamics and verified them in rodent seizure models. Finally, a computational model was developed, successfully explaining how the complex spatiotemporal evolution of focal seizures emerges from simple neurophysiological principles. In Chapter 1, the long-standing behavioral manifestations and the most up-to-date electrophysiology findings are reviewed. This section details the inspiration for the studies reported in the subsequent chapters. In Chapter 2, I describe several statistical methods for estimating traveling wave velocities. I show most ictal discharges can be described as traveling waves whose velocities contain rich information about the stages of seizure evolution. I compare performance of various statistical methods and propose a robust approach to boost the quality of each method’s estimation results. In Chapter 3, I show how inhibition modulates seizure propagation patterns. Surround inhibition spatially restrains focal seizures and masks excitatory projections of ictal activities. When compromised, two patterns of seizure propagation emerge according to the position of inhibition defects relative to the ictal focus. I show that two distant ictal foci can communicate via physiological connectivity without any chronic rewiring processes – confirming the existence of long-range propagation pathways that could lead to epileptic network formation. In Chapter 4, I show that thalamic inputs might be necessary for interictal epileptiform discharges (IEDs). The relative positions between IEDs and ictal foci indicate that surround inhibition, shown in the previous chapter, can be exhausted by repetitive exposure to ictal projections. In Chapter 5, I propose a neural network model that can explain both long-standing behavioral observations of seizures and account for the most up-to-date electrophysiological recordings of spontaneous human focal seizures. The model relies on few assumptions, all of which are proved or supported in earlier chapters of this thesis. The model explains phasic evolution of seizure dynamics – how the commonly observed patterns arise from simple neurophysiological principles, as well as seizure onset subtypes, traveling wave directions and speeds. The model also predicts how spontaneous seizures might arise from synaptic plasticity. The chapter ends with a discussion of the model’s implications and future work. The thesis is organized in a way that each chapter can be read independently, with Chapter 5 summarizing the central theory spanning the whole study. Each chapter is also tightly linked to a clinically relevant question. In sum, the dissertation’s goal is to provide an in-principle understanding of focal seizure dynamics. With rapid advancement of clinical and experimental tools, I believe this work provides a roadmap for future therapies for epilepsy patients

Capture of fixation by rotational flow; a deterministic hypothesis regarding scaling and stochasticity in fixational eye movements.

Visual scan paths exhibit complex, stochastic dynamics. Even during visual fixation, the eye is in constant motion. Fixational drift and tremor are thought to reflect fluctuations in the persistent neural activity of neural integrators in the oculomotor brainstem, which integrate sequences of transient saccadic velocity signals into a short term memory of eye position. Despite intensive research and much progress, the precise mechanisms by which oculomotor posture is maintained remain elusive. Drift exhibits a stochastic statistical profile which has been modeled using random walk formalisms. Tremor is widely dismissed as noise. Here we focus on the dynamical profile of fixational tremor, and argue that tremor may be a signal which usefully reflects the workings of oculomotor postural control. We identify signatures reminiscent of a certain flavor of transient neurodynamics; toric traveling waves which rotate around a central phase singularity. Spiral waves play an organizational role in dynamical systems at many scales throughout nature, though their potential functional role in brain activity remains a matter of educated speculation. Spiral waves have a repertoire of functionally interesting dynamical properties, including persistence, which suggest that they could in theory contribute to persistent neural activity in the oculomotor postural control system. Whilst speculative, the singularity hypothesis of oculomotor postural control implies testable predictions, and could provide the beginnings of an integrated dynamical framework for eye movements across scales

The complexity of dynamics in small neural circuits

Mean-field theory is a powerful tool for studying large neural networks. However, when the system is composed of a few neurons, macroscopic differences between the mean-field approximation and the real behavior of the network can arise. Here we introduce a study of the dynamics of a small firing-rate network with excitatory and inhibitory populations, in terms of local and global bifurcations of the neural activity. Our approach is analytically tractable in many respects, and sheds new light on the finite-size effects of the system. In particular, we focus on the formation of multiple branching solutions of the neural equations through spontaneous symmetry-breaking, since this phenomenon increases considerably the complexity of the dynamical behavior of the network. For these reasons, branching points may reveal important mechanisms through which neurons interact and process information, which are not accounted for by the mean-field approximation.Comment: 34 pages, 11 figures. Supplementary materials added, colors of figures 8 and 9 fixed, results unchange

Dynamics and precursor signs for phase transitions in neural systems

This thesis investigates neural state transitions associated with sleep, seizure and anaesthesia. The aim is to address the question: How does a brain traverse the critical threshold between distinct cortical states, both healthy and pathological? Specifically we are interested in sub-threshold neural behaviour immediately prior to state transition. We use theoretical neural modelling (single spiking neurons, a network of these, and a mean-field continuum limit) and in vitro experiments to address this question. Dynamically realistic equations of motion for thalamic relay neuron, reticular nuclei, cortical pyramidal and cortical interneuron in different vigilance states are developed, based on the Izhikevich spiking neuron model. A network of cortical neurons is assembled to examine the behaviour of the gamma-producing cortical network and its transition to lower frequencies due to effect of anaesthesia. Then a three-neuron model for the thalamocortical loop for sleep spindles is presented. Numerical simulations of these networks confirms spiking consistent with reported in vivo measurement results, and provides supporting evidence for precursor indicators of imminent phase transition due to occurrence of individual spindles. To complement the spiking neuron networks, we study the Wilson–Cowan neural mass equations describing homogeneous cortical columns and a 1D spatial cluster of such columns. The abstract representation of cortical tissue by a pair of coupled integro-differential equations permits thorough linear stability, phase plane and bifurcation analyses. This model shows a rich set of spatial and temporal bifurcations marking the boundary to state transitions: saddle-node, Hopf, Turing, and mixed Hopf–Turing. Close to state transition, white-noise-induced subthreshold fluctuations show clear signs of critical slowing down with prolongation and strengthening of autocorrelations, both in time and space, irrespective of bifurcation type. Attempts at in vitro capture of these predicted leading indicators form the last part of the thesis. We recorded local field potentials (LFPs) from cortical and hippocampal slices of mouse brain. State transition is marked by the emergence and cessation of spontaneous seizure-like events (SLEs) induced by bathing the slices in an artificial cerebral spinal fluid containing no magnesium ions. Phase-plane analysis of the LFP time-series suggests that distinct bifurcation classes can be responsible for state change to seizure. Increased variance and growth of spectral power at low frequencies (f < 15 Hz) was observed in LFP recordings prior to initiation of some SLEs. In addition we demonstrated prolongation of electrically evoked potentials in cortical tissue, while forwarding the slice to a seizing regime. The results offer the possibility of capturing leading temporal indicators prior to seizure generation, with potential consequences for understanding epileptogenesis. Guided by dynamical systems theory this thesis captures evidence for precursor signs of phase transitions in neural systems using mathematical and computer-based modelling as well as in vitro experiments

Birth and destruction of collective oscillations in a network of two populations of coupled type 1 neurons

We study the macroscopic dynamics of large networks of excitable type 1 neurons composed of two populations interacting with disparate but symmetric intra- and inter-population coupling strengths. This nonuniform coupling scheme facilitates symmetric equilibria, where both populations display identical firing activity, characterized by either quiescent or spiking behavior, or asymmetric equilibria, where the firing activity of one population exhibits quiescent but the other exhibits spiking behavior. Oscillations in the firing rate are possible if neurons emit pulses with non-zero width but are otherwise quenched. Here, we explore how collective oscillations emerge for two statistically identical neuron populations in the limit of an infinite number of neurons. A detailed analysis reveals how collective oscillations are born and destroyed in various bifurcation scenarios and how they are organized around higher codimension bifurcation points. Since both symmetric and asymmetric equilibria display bistable behavior, a large configuration space with steady and oscillatory behavior is available. Switching between configurations of neural activity is relevant in functional processes such as working memory and the onset of collective oscillations in motor control