arXiv subjects

Sue Ann Campbell

Publications and source records attributed to Sue Ann Campbell.

18 recordsLinked to original sources

arXiv2026

Stability of Phase-locked States of Weakly Coupled Izhikevich Neurons

The Izhikevich model is a computationally efficient neuron model that can exhibit a wide range of firing patterns observed in the brain. However, it is a discontinuous dynamical system, which means the methods for applying weakly coupled oscillator theory developed for continuous dynamical systems cannot be applied. Therefore, the collective behaviour of coupled Izhikevich model has not been fully studied. To our knowledge, we carry out the first computation of the phase model for an Izhikevich neuronal model with chemical synapses. Using the phase model we study the existence and stability of phase-locking states and how these vary with parameters. Moreover, we find an interesting connection between the stability of anti-phase states of the phase model and the bifurcation behaviour of the uncoupled Izhikevich model itself. The accuracy of the prediction of the phase model is supported by simulations of two weakly coupled Izhikevich neurons.

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arXiv2026

Phase model analysis of the effect of M-current on neural synchrony in hippocampal networks

Neural assemblies, transiently coordinated groups of neurons, observed in the hippocampus are thought to underlie the formation of episodic memories. Acetylcholine (ACh), a neuromodulator, that is received by the hippocampus, plays a critical role in memory and learning. A well supported hypothesis suggests that high levels of ACh during active exploration and rapid eye movement (REM) sleep promote memory encoding, while low levels during quiet waking and slow-wave sleep (SWS) support memory consolidation. We study this bidirectional role of ACh in neural assembly formation through its effect on the synchrony among neurons. We consider a network model of pyramidal neurons, each equipped with a slow, voltage-dependent, non-inactivating potassium current (M-current), which is downregulated in the presence of ACh. Neural assemblies are represented as cluster solutions to this system. Using a one-dimensional phase model reduction of a pair of weakly coupled pyramidal neurons under different levels of the M-current, we predict the symmetric cluster solutions that may emerge in larger networks equipped with all-to-all globally homogeneous, symmetric distance-dependent and nearest-neighbours coupling architectures. We find that under low ACh conditions, the network can fully synchronize, whereas high levels can desynchronize the network into multiple stable symmetric cluster solutions representing distinct neural assemblies.

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arXiv2026

Approximate Macroscopic Dynamics of Spiking Neural Networks Based on Solutions to the Transport Equation

Firing rate fluctuations in neural populations are observed experimentally over multiple time scales, in single neurons, across trials when elicited by stimuli, and across populations. In this work, we examine how firing rate fluctuations emerge in networks of coupled integrate-and-fire neurons as a function of the initial distribution of voltages in networks with time-varying inputs. We analytically derive an approximation for the evolution of the instantaneous population rate or flux as a function of the initial voltage distribution through a Fokker-Planck system. Unlike earlier mean field approaches based on asynchronous or constant flux steady state solutions to the Fokker-Planck system, the approach considered here is based on the transport solution to the advection equation and assumes that the time-varying inputs are slow, and the neurons are in the excitation-driven regime. The transport mean field system predicts how firing rate fluctuations emerge from a dynamic interaction between time-varying inputs, initial densities, and coupling in populations of neurons.

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arXiv2025

Influence of Large Mean Delay on Distributed Delay Differential Equations Dynamics: Application to a Neural Mass Model

Delay differential equations (DDEs) with large delays play a pivotal role in understanding stability and bifurcations in systems ranging from neural networks to laser dynamics. While prior work has extensively studied DDEs with discrete delays, the impact of distributed delays has been less explored. This paper investigates the spectrum of linear DDEs with a uniformly distributed delay kernel, with mean delay $τ_m$ and a half-width of $ρ$. When $τ_m\to\infty$, we carry out asymptotic analysis and show that the spectrum splits into (i) a strong critical spectrum referring to a finite set of isolated, pure imaginary eigenvalues that are unaffected by delay, (ii) an asymptotic strong spectrum consisting of a finite set of eigenvalues with limits that are determined by non-delayed terms in the model and (iii) a pseudo-continuous spectrum consisting of infinitely many eigenvalues that limit on the imaginary axis, with real parts that scale linearly with the delay. Although this behavior is similar to the fixed delay case, the distributed delay introduces additional spectral features, including an infinite countable number of horizontal asymptotes in the pseudo-continuous spectrum at frequencies $ω= kπ/ρ$, where $k\in \mathbb{Z}\setminus\{0\}$. We validate our theoretical result through several examples and compare our findings with fixed-delay results from the literature. Finally, we apply the results to study the stability and bifurcations of a Wilson-Cowan model with a delayed self-coupling, large mean delay, and homeostatic plasticity.

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arXiv2023

Distributed Delay and Desynchronization in a Brain Network Model

We consider a neural field model which consists of a network of an arbitrary number of Wilson-Cowan nodes with homeostatic adjustment of the inhibitory coupling strength and time delayed, excitatory coupling. We extend previous work on this model to include distributed time delays with commonly used kernel distributions: delta function, uniform distribution and gamma distribution. Focussing on networks which satisfy a constant row sum condition, we show how each eigenvalue of the connectivity matrix may be related to a Hopf bifurcation and that the eigenvalue determines whether the bifurcation leads to synchronized or desynchronized oscillatory behaviour. We consider two example networks, one with all real eigenvalues (bi-directional ring) and one with some complex eigenvalues (uni-directional ring). In bi-directional rings, the Hopf curves are organized so that only the synchronized Hopf leads to asymptotically stable behaviour. Thus the behaviour in the network is always synchronous. In the uni-directional ring networks, however, intersection points of asynchronous and synchronous Hopf curves may occur resulting in double Hopf bifurcation points. Thus asymptotically stable synchronous and asynchronous limit cycles can occur as well as torus-like solutions which combine synchronous and asynchronous behaviour. Increasing the size of the network or the mean time delay makes these intersection points, and the associated asynchronous behaviour, more likely to occur. Numerical approaches are used to confirm the findings, with Hopf bifurcation curves plotted using Wolfram Mathematica. These insights offer a deeper understanding of the mechanisms underlying desynchronization in large networks of oscillators.

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arXiv2023

Synaptic delay induced macroscopic dynamics of the large-scale network of Izhikevich neurons

We consider a large network of Izhikevich neurons. Each neuron has a quadratic integrate-and-fire type model with a recovery variable modelling spike frequency adaptation (SFA). We introduce a biologically motivated synaptic current expression and a delay in the synaptic transmission. Following the Ott-Antonsen theory, we reduce the network model to a mean-field system of delayed differential equations. Numerical bifurcation analysis allows us to locate higher-codimension bifurcations and to identify the regions in the parameter space where the network exhibits changes in the macroscopic dynamics, including transitions between states where the individual neurons exhibit asynchronous tonic firing and different types of synchronous bursting. We investigate the impact of the heterogeneity of the quenched input current, the SFA mechanism and the synaptic delay on macroscopic dynamics. In the limit that the heterogeneity goes to zero, our perturbation and bifurcation analysis shows that the behaviour of the mean-field model remains consistent, although this limit breaks an assumption of the model reduction. For a single population of neurons with SFA, the synaptic delay has little effect on the generation of COs for weak coupling, but favours their emergence beyond that, and even induces new macroscopic dynamics. In particular, Torus bifurcations may occur, and these are a crucial mechanism for the emergence of population bursting with two nested frequencies. We discuss how these solutions may relate to cross-frequency coupling which is potentially relevant for understanding healthy and pathological brain functions.

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arXiv2022

Exact mean-field models for spiking neural networks with adaptation

Networks of spiking neurons with adaption have been shown to be able to reproduce a wide range of neural activities, including the emergent population bursting and spike synchrony that underpin brain disorders and normal function. Exact mean-field models derived from spiking neural networks are extremely valuable, as such models can be used to determine how individual neuron and network parameters interact to produce macroscopic network behaviour. In the paper, we derive and analyze a set of exact mean-field equations for the neural network with spike frequency adaptation. Specifically, our model is a network of Izhikevich neurons, where each neuron is modeled by a two dimensional system consisting of a quadratic integrate and fire equation plus an equation which implements spike frequency adaptation. Previous work deriving a mean-field model for this type of network, relied on the assumption of sufficiently slow dynamics of the adaptation variable. However, this approximation did not succeeded in establishing an exact correspondence between the macroscopic description and the realistic neural network, especially when the adaptation time constant was not large. The challenge lies in how to achieve a closed set of mean-field equations with the inclusion of the mean-field expression of the adaptation variable. We address this challenge by using a Lorentzian ansatz combined with the moment closure approach to arrive at the mean-field system in the thermodynamic limit. The resulting macroscopic description is capable of qualitatively and quantitatively describing the collective dynamics of the neural network, including transition between tonic firing and bursting.

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arXiv2021

Hysteresis bifurcation and application to delayed Fitzhugh-Nagumo neural systems

Hysteresis dynamics has been described in a vast number of biological experimental studies. Many such studies are phenomenological and a mathematical appreciation has not attracted enough attention. In the paper, we explore the nature of hysteresis and study it from the dynamical system point of view by using the bifurcation and perturbation theories. We firstly make a classification of hysteresis according to the system behaviours transiting between different types of attractors. Then, we focus on a mathematically amenable situation where hysteretic movements between the equilibrium point and the limit cycle are initiated by a subcritical Hopf bifurcation and a saddle-node bifurcation of limit cycles. We present a analytical framework by using the method of multiple scales to obtain the normal form up to the fifth order. Theoretical results are compared with time domain simulations and numerical continuation, showing good agreement. Although we consider the time-delayed FitzHugh-Nagumo neural system in the paper, the generalization should be clear to other systems or parameters. The general framework we present in the paper can be naturally extended to the notion of bursting activity in neuroscience where hysteresis is a dominant mechanism to generate bursting oscillations.

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arXiv2021

M-current Induced Bogdanov-Takens Bifurcation and Switching of Neuron Excitability Class

In this work, we consider a general conductance-based neuron model with the inclusion of the acetycholine sensitive, M-current. We study bifurcations in the parameter space consisting of the applied current, $I_{app}$ the maximal conductance of the M-current, $g_M$, and the conductance of the leak current, $g_L$. We give precise conditions for the model that ensure the existence of a Bogdanov-Takens (BT) point and show such a point can occur by varying $I_{app}$ and $g_{M}$. We discuss the case when the BT point becomes a Bogdanov-Takens-Cusp (BTC) point and show that such a point can occur in the three dimensional parameter space. The results of the bifurcation analysis are applied to different neuronal models and are verified and supplemented by numerical bifurcation diagrams generated using the package MATCONT. We conclude that there is a transition in the neuronal excitability type organized by the BT point and the neuron switches from Class-I to Class-II as conductance of the M-current increases.

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arXiv2020

Normalized Connectomes Show Increased Synchronizability with Age Through Their Second Largest Eigenvalue

The synchronization of different brain regions is widely observed under both normal and pathological conditions such as epilepsy. However, the relationship between the dynamics of these brain regions, the connectivity between them, and the ability to synchronize remains an open question. We investigated the problem of inter-region synchronization in networks of Wilson-Cowan/Neural field equations with homeostatic plasticity, each of which acts as a model for an isolated brain region. We considered arbitrary connection profiles with only one constraint: the rows of the connection matrices are all identically normalized. We found that these systems often synchronize to the solution obtained from a single, self-coupled neural region. We analyze the stability of this solution through a straightforward modification of the Master Stability Function (MSF) approach and found that synchronized solutions lose stability for connectivity matrices when the second largest positive eigenvalue is sufficiently large, for values of the global coupling parameter that are not too large. This result was numerically confirmed for ring systems and lattices and was also robust to small amounts of heterogeneity in the homeostatic set points in each node. Finally, we tested this result on connectomes obtained from 196 subjects over a broad age range (4-85 years) from the Human Connectome Project. We found that the second largest eigenvalue tended to decrease with age, indicating an increase in synchronizability that may be related to the increased prevalence of epilepsy with old age.

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arXiv2020

A Phase Model with Large Time Delayed Coupling

We consider two identical oscillators with weak, time delayed coupling. We start with a general system of delay differential equations then reduce it to a phase model. With the assumption of large time delay, the resulting phase model has an explicit delay and phase shift in the argument of the phases and connection function, respectively. Using the phase model, we prove that for any type of oscillators and any coupling, the in-phase and anti-phase phase-locked solutions always exist and give conditions for their stability. We show that for small delay these solutions are unique, but with large enough delay multiple solutions of each type with different frequencies may occur. We give conditions for the existence and stability of other types of phase-locked solutions. We discuss the various bifurcations that can occur in the phase model as the time delay is varied. The results of the phase model analysis are applied to Morris-Lecar oscillators with diffusive coupling and compared with numerical studies of the full system of delay differential equations. We also consider the case of small time delay and compare the results with the existing ones in the literature.

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arXiv2018

Chaos in Homeostatically Regulated Neural Systems

Low-dimensional yet rich dynamics often emerge in the brain. Examples include oscillations and chaotic dynamics during sleep, epilepsy, and voluntary movement. However, a general mechanism for the emergence of low dimensional dynamics remains elusive. Here, we consider Wilson-Cowan networks and demonstrate through numerical and analytical work that a type of homeostatic regulation of the network firing rates can paradoxically lead to a rich dynamical repertoire. The dynamics include mixed-mode oscillations, mixed-mode chaos, and chaotic synchronization. This is true for single recurrently coupled node, pairs of reciprocally coupled nodes without self-coupling, and networks coupled through experimentally determined weights derived from functional magnetic resonance imaging data. In all cases, the stability of the homeostatic set point is analytically determined or approximated. The dynamics at the network level are directly determined by the behavior of a single node system through synchronization in both oscillatory and non-oscillatory states. Our results demonstrate that rich dynamics can be preserved under homeostatic regulation or even be caused by homeostatic regulation.

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arXiv2017

Geometric Analysis of Synchronization in Neuronal Networks with Global Inhibition and Coupling Delays

We study synaptically coupled neuronal networks to identify the role of coupling delays in network's synchronized behaviors. We consider a network of excitable, relaxation oscillator neurons where two distinct populations, one excitatory and one inhibitory, are coupled and interact with each other. The excitatory population is uncoupled, while the inhibitory population is tightly coupled. A geometric singular perturbation analysis yields existence and stability conditions for synchronization states under different firing patterns between the two populations, along with formulas for the periods of such synchronous solutions. Our results demonstrate that the presence of coupling delays in the network promotes synchronization. Numerical simulations are conducted to supplement and validate analytical results. We show the results carry over to a model for spindle sleep rhythms in thalamocortical networks, one of the biological systems which motivated our study. The analysis helps to explain how coupling delays in either excitatory or inhibitory synapses contribute to producing synchronized rhythms.

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arXiv2017

Stability in a scalar differential equation with multiple, distributed time delays

We consider a linear scalar delay differential equation (DDE), consisting of two arbitrary distributed time delays. We formulate necessary conditions for stability of the trivial solution which are independent of the distributions. For the case of one discrete and one gamma distributed delay, we give an explicit description of the region of stability of the trivial solution and discuss how this depends on the model parameters.

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arXiv2016

Phase models and clustering in networks of oscillators with delayed coupling

We consider a general model for a network of oscillators with time delayed, circulant coupling. We use the theory of weakly coupled oscillators to reduce the system of delay differential equations to a phase model where the time delay enters as a phase shift. We use the phase model to study the existence and stability of cluster solutions. Cluster solutions are phase locked solutions where the oscillators separate into groups. Oscillators within a group are synchronized while those in different groups are phase-locked. We give model independent existence and stability results for symmetric cluster solutions. We show that the presence of the time delay can lead to the coexistence of multiple stable clustering solutions. We apply our analytical results to a network of Morris Lecar neurons and compare these results with numerical continuation and simulation studies.

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arXiv2014

One-Dimensional Population Density Approaches to Recurrently Coupled Networks of Neurons with Noise

Mean-field systems have been previously derived for networks of coupled, two-dimensional, integrate-and-fire neurons such as the Izhikevich, adapting exponential (AdEx) and quartic integrate and fire (QIF), among others. Unfortunately, the mean-field systems have a degree of frequency error and the networks analyzed often do not include noise when there is adaptation. Here, we derive a one-dimensional partial differential equation (PDE) approximation for the marginal voltage density under a first order moment closure for coupled networks of integrate-and-fire neurons with white noise inputs. The PDE has substantially less frequency error than the mean-field system, and provides a great deal more information, at the cost of analytical tractability. The convergence properties of the mean-field system in the low noise limit are elucidated. A novel method for the analysis of the stability of the asynchronous tonic firing solution is also presented and implemented. Unlike previous attempts at stability analysis with these network types, information about the marginal densities of the adaptation variables is used. This method can in principle be applied to other systems with nonlinear partial differential equations.

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arXiv2014

Non-smooth Bifurcations of Mean Field Systems of Two-Dimensional Integrate and Fire Neurons

Mean-field systems have been recently derived that adequately predict the behaviors of large networks of coupled integrate-and-fire neurons [14]. The mean-field system for a network of neurons with spike frequency adaptation is typically a pair of differential equations for the mean adaptation and mean synaptic gating variable of the network. These differential equations are non-smooth, and in particular are piecewise smooth continuous (PWSC). Here, we analyze the smooth and non-smooth bifurcation structure of these equations and show that the system is organized around a pair of co-dimension two bifurcations that involve, respectively, the collision between a Hopf equilibrium point and a switching manifold, and a saddle-node equilibrium point and a switching manifold. These two co-dimension 2 bifurcations can coalesce into a co-dimension 3 non-smooth bifurcation. As the mean-field system we study is a non-generic piecewise smooth continuous system, we discuss possible regularizations of this system and how the bifurcations which occur are related to non-smooth bifurcations displayed by generic PWSC systems.

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arXiv2013

Mean-Field Models for Heterogeneous Networks of Two-Dimensional Integrate and Fire Models

We analytically derive mean-field models for all-to-all coupled networks of heterogeneous, adapting, two-dimensional integrate and fire neurons. The class of models we consider includes the Izhikevich, adaptive exponential, and quartic integrate and fire models. The heterogeneity in the parameters leads to different moment closure assumptions that can be made in the derivation of the mean-field model from the population density equation for the large network. Three different moment closure assumptions lead to three different mean-field systems. These systems can be used for distinct purposes such as bifurcation analysis of the large networks, prediction of steady state firing rate distributions, parameter estimation for actual neurons, and faster exploration of the parameter space. We use the mean-field systems to analyze adaptation induced bursting under realistic sources of heterogeneity in multiple parameters. Our analysis demonstrates that the presence of heterogeneity causes the Hopf bifurcation associated with the emergence of bursting to change from sub-critical to super-critical. This is confirmed with numerical simulations of the full network for biologically reasonable parameter values. This change decreases the plausibility of adaptation being the cause of bursting in hippocampal area CA3, an area with a sizable population of heavily coupled, strongly adapting neurons.

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