Codimension-two homoclinic bifurcations underlying spike adding in the Hindmarsh-Rose burster

The well-studied Hindmarsh-Rose model of neural action potential is revisited from the point of view of global bifurcation analysis. This slow-fast system of three paremeterised differential equations is arguably the simplest reduction of Hodgkin-Huxley models capable of exhibiting all qualitatively important distinct kinds of spiking and bursting behaviour. First, keeping the singular perturbation parameter fixed, a comprehensive two-parameter bifurcation diagram is computed by brute force. Of particular concern is the parameter regime where lobe-shaped regions of irregular bursting undergo a transition to stripe-shaped regions of periodic bursting. The boundary of each stripe represents a fold bifurcation that causes a smooth spike-adding transition where the number of spikes in each burst is increased by one. Next, numerical continuation studies reveal that the global structure is organised by various curves of homoclinic bifurcations. In particular the lobe to stripe transition is organised by a sequence of codimension-two orbit- and inclination-flip points that occur along {\em each} homoclinic branch. Each branch undergoes a sharp turning point and hence approximately has a double-cover of the same curve in parameter space. The sharp turn is explained in terms of the interaction between a two-dimensional unstable manifold and a one-dimensional slow manifold in the singular limit. Finally, a new local analysis is undertaken using approximate Poincar\'{e} maps to show that the turning point on each homoclinic branch in turn induces an inclination flip that gives birth to the fold curve that organises the spike-adding transition. Implications of this mechanism for explaining spike-adding behaviour in other excitable systems are discussed.Comment: 32 pages, 18 figures, submitted to SIAM Journal on Applied Dynamical System

Spike-adding and reset-induced canard cycles in adaptive integrate and fire models

We study a class of planar integrate and fire (IF) models called adaptive integrate and fire (AIF) models, which possesses an adaptation vari- able on top of membrane potential, and whose subthreshold dynamics is piece- wise linear (PWL). These AIF models therefore have two reset conditions, which enable bursting dynamics to emerge for suitable parameter values. Such models can be thought of as hybrid dynamical systems. We consider a par- ticular slow dynamics within AIF models and prove the existence of bursting cycles with N resets, for any integer N. Furthermore, we study the transition between N- and (N + 1)-reset cycles upon vanishingly small parameter vari- ations and prove (for N = 2) that such transitions are organised by canard cycles. Finally, using numerical continuation we compute branches of bursting cycles, including canard-explosive branches, in these AIF models, by suitably recasting the periodic problem as a two-point boundary-value problem

Effect of the electromagnetic induction on a modified memristive neural map model

The significance of discrete neural models lies in their mathematical simplicity and computational ease. This research focuses on enhancing a neural map model by incorporating a hyperbolic tangent-based memristor. The study extensively explores the impact of magnetic induction strength on the model's dynamics, analyzing bifurcation diagrams and the presence of multistability. Moreover, the investigation extends to the collective behavior of coupled memristive neural maps with electrical, chemical, and magnetic connections. The synchronization of these coupled memristive maps is examined, revealing that chemical coupling exhibits a broader synchronization area. Additionally, diverse chimera states and cluster synchronized states are identified and discussed

Complex oscillations with multiple timescales - Application to neuronal dynamics

The results gathered in this thesis deal with multiple time scale dynamical systems near non-hyperbolic points, giving rise to canard-type solutions, in systems of dimension 2, 3 and 4. Bifurcation theory and numerical continuation methods adapted for such systems are used to analyse canard cycles as well as canard-induced complex oscillations in three-dimensional systems. Two families of such complex oscillations are considered: mixed-mode oscillations (MMOs) in systems with two slow variables, and bursting oscillations in systems with two fast variables. In the last chapter, we present recent results on systems with two slow and two fast variables, where both MMO-type dynamics and bursting-type dynamics can arise and where complex oscillations are also organised by canard solutions. The main application area that we consider here is that of neuroscience, more precisely low-dimensional point models of neurons displaying both sub-threshold and spiking behaviour. We focus on analysing how canard objects allow to control the oscillatory patterns observed in these neuron models, in particular the crossings of excitability thresholds

Six Types of Multistability in a Neuronal Model Based on Slow Calcium Current

Background: Multistability of oscillatory and silent regimes is a ubiquitous phenomenon exhibited by excitable systems such as neurons and cardiac cells. Multistability can play functional roles in short-term memory and maintaining posture. It seems to pose an evolutionary advantage for neurons which are part of multifunctional Central Pattern Generators to possess multistability. The mechanisms supporting multistability of bursting regimes are not well understood or classified. Methodology/Principal Findings: Our study is focused on determining the bio-physical mechanisms underlying different types of co-existence of the oscillatory and silent regimes observed in a neuronal model. We develop a low-dimensional model typifying the dynamics of a single leech heart interneuron. We carry out a bifurcation analysis of the model and show that it possesses six different types of multistability of dynamical regimes. These types are the co-existence of 1) bursting and silence, 2) tonic spiking and silence, 3) tonic spiking and subthreshold oscillations, 4) bursting and subthreshold oscillations, 5) bursting, subthreshold oscillations and silence, and 6) bursting and tonic spiking. These first five types of multistability occur due to the presence of a separating regime that is either a saddle periodic orbit or a saddle equilibrium. We found that the parameter range wherein multistability is observed is limited by the parameter values at which the separating regimes emerge and terminate. Conclusions: We developed a neuronal model which exhibits a rich variety of different types of multistability. We described a novel mechanism supporting the bistability of bursting and silence. This neuronal model provides a unique opportunity to study the dynamics of networks with neurons possessing different types of multistability

Low-dimensional models of single neurons: A review

The classical Hodgkin-Huxley (HH) point-neuron model of action potential generation is four-dimensional. It consists of four ordinary differential equations describing the dynamics of the membrane potential and three gating variables associated to a transient sodium and a delayed-rectifier potassium ionic currents. Conductance-based models of HH type are higher-dimensional extensions of the classical HH model. They include a number of supplementary state variables associated with other ionic current types, and are able to describe additional phenomena such as sub-threshold oscillations, mixed-mode oscillations (subthreshold oscillations interspersed with spikes), clustering and bursting. In this manuscript we discuss biophysically plausible and phenomenological reduced models that preserve the biophysical and/or dynamic description of models of HH type and the ability to produce complex phenomena, but the number of effective dimensions (state variables) is lower. We describe several representative models. We also describe systematic and heuristic methods of deriving reduced models from models of HH type

Bifurcations of piecewise smooth flows:perspectives, methodologies and open problems

In this paper, the theory of bifurcations in piecewise smooth flows is critically surveyed. The focus is on results that hold in arbitrarily (but finitely) many dimensions, highlighting significant areas where a detailed understanding is presently lacking. The clearest results to date concern equilibria undergoing bifurcations at switching boundaries, and limit cycles undergoing grazing and sliding bifurcations. After discussing fundamental concepts, such as topological equivalence of two piecewise smooth systems, discontinuity-induced bifurcations are defined for equilibria and limit cycles. Conditions for equilibria to exist in n-dimensions are given, followed by the conditions under which they generically undergo codimension-one bifurcations. The extent of knowledge of their unfoldings is also summarized. Codimension-one bifurcations of limit cycles and boundary-intersection crossing are described together with techniques for their classification. Codimension-two bifurcations are discussed with suggestions for further study

Digital Implementation of Bio-Inspired Spiking Neuronal Networks

Spiking Neural Network as the third generation of artificial neural networks offers a promising solution for future computing, prosthesis, robotic and image processing applications. This thesis introduces digital designs and implementations of building blocks of a Spiking Neural Networks (SNNs) including neurons, learning rule, and small networks of neurons in the form of a Central Pattern Generator (CPG) which can be used as a module in control part of a bio-inspired robot. The circuits have been developed using Verilog Hardware Description Language (VHDL) and simulated through Modelsim and compiled and synthesised by Altera Qurtus Prime software for FPGA devices. Astrocyte as one of the brain cells controls synaptic activity between neurons by providing feedback to neurons. A novel digital hardware is proposed for neuron-synapseastrocyte network based on the biological Adaptive Exponential (AdEx) neuron and Postnov astrocyte cell model. The network can be used for implementation of large scale spiking neural networks. Synthesis of the designed circuits shows that the designed astrocyte circuit is able to imitate its biological model and regulate the synapse transmission, successfully. In addition, synthesis results confirms that the proposed design uses less than 1% of available resources of a VIRTEX II FPGA which saves up to 4.4% of FPGA resources in comparison to other designs. Learning rule is an essential part of every neural network including SNN. In an SNN, a special type of learning called Spike Timing Dependent Plasticity (STDP) is used to modify the connection strength between the spiking neurons. A pair-based STDP (PSTDP) works on pairs of spikes while a Triplet-based STDP (TSTDP) works on triplets of spikes to modify the synaptic weights. A low cost, accurate, and configurable digital architectures are proposed for PSTDP and TSTDP learning models. The proposed circuits have been compared with the state of the art methods like Lookup Table (LUT), and Piecewise Linear approximation (PWL). The circuits can be employed in a large-scale SNN implementation due to their compactness and configurability. Most of the neuron models represented in the literature are introduced to model the behavior of a single neuron. Since there is a large number of neurons in the brain, a population-based model can be helpful in better understanding of the brain functionality, implementing cognitive tasks and studying the brain diseases. Gaussian Wilson-Cowan model as one of the population-based models represents neuronal activity in the neocortex region of the brain. A digital model is proposed for the GaussianWilson-Cowan and examined in terms of dynamical and timing behavior. The evaluation indicates that the proposed model is able to generate the dynamical behavior as the original model is capable of. Digital architectures are implemented on an Altera FPGA board. Experimental results show that the proposed circuits take maximum 2% of the resources of a Stratix Altera board. In addition, static timing analysis indicates that the circuits can work in a maximum frequency of 244 MHz

Chimera states in multi-strain epidemic models with temporary immunity

We investigate a time-delayed epidemic model for multi-strain diseases with temporary immunity. In the absence of cross-immunity between strains, dynamics of each individual strain exhibits emergence and annihilation of limit cycles due to a Hopf bifurcation of the endemic equilibrium, and a saddle-node bifurcation of limit cycles depending on the time delay associated with duration of temporary immunity. Effects of all-to-all and non-local coupling topologies are systematically investigated by means of numerical simulations, and they suggest that cross-immunity is able to induce a diverse range of complex dynamical behaviors and synchronization patterns, including discrete traveling waves, solitary states, and amplitude chimeras. Interestingly, chimera states are observed for narrower cross-immunity kernels, which can have profound implications for understanding the dynamics of multi-strain diseases