Inverse Stochastic Resonance (ISR) is a phenomenon in which the average spiking rate of a neuron exhibits a minimum with respect to noise. ISR has been studied in individual neurons, but here, we investigate ISR in scale-free networks, where the average spiking rate is calculated over the neuronal population. We use Hodgkin-Huxley model neurons with channel noise (i.e., stochastic gating variable dynamics), and the network connectivity is implemented via electrical or chemical connections (i.e., gap junctions or excitatory/inhibitory synapses). We find that the emergence of ISR depends on the interplay between each neuron's intrinsic dynamical structure, channel noise, and network inputs, where the latter in turn depend on network structure parameters. We observe that with weak gap junction or excitatory synaptic coupling, network heterogeneity and sparseness tend to favor the emergence of ISR. With inhibitory coupling, ISR is quite robust. We also identify dynamical mechanisms that underlie various features of this ISR behavior. Our results suggest possible ways of experimentally observing ISR in actual neuronal systems.

PLoS Computational Biology 13(7): e1005646 (2017). Journal link.We investigate the behavior of a model neuron that receives a biophysically-realistic noisy post-synaptic current based on uncorrelated spiking activity from a large number of afferents. We show that, with static synapses, such noise can give rise to inverse stochastic resonance (ISR) as a function of the presynaptic firing rate. We compare this to the case with dynamic synapses that feature short-term synaptic plasticity, and show that the interval of presynaptic firing rate over which ISR exists can be extended or diminished. We consider both short-term depression and facilitation. Interestingly, we find that a double inverse stochastic resonance (DISR), with two distinct wells centered at different presynaptic firing rates, can appear.

Physical Review E 95, 012404 (2017). Journal linkAn externally-applied electric field can polarize a neuron, especially a neuron with elongated dendrites, and thus modify its excitability. Here we use a computational model to examine, predict, and explain these effects. We use a two-compartment Pinsky-Rinzel model neuron polarized by an electric potential difference imposed between its compartments, and we apply an injected ramp current. We vary three model parameters: the magnitude of the applied potential difference, the extracellular potassium concentration, and the rate of current injection. A study of the Time-To-First-Spike (TTFS) as a function of polarization leads to the identification of three regions of polarization strength that have di erent effects. In the weak region, the TTFS increases linearly with polarization. In the intermediate region, the TTFS increases either sub- or super-linearly, depending on the current injection rate and the extracellular potassium concentration. In the strong region, the TTFS decreases. Our results in the weak and strong region are consistent with experimental observations, and in the intermediate region, we predict novel e ects that depend on experimentally-accessible parameters. We find that active channels in the dendrite play a key role in these effects. Our qualitative results were found to be robust over a wide range of inter-compartment conductances and the ratio of somatic to dendritic membrane areas. In addition, we discuss preliminary results where synaptic inputs replace the ramp injection protocol. The insights and conclusions were found to extend from our polarized PR model to a polarized PR model with Ih dendritic currents. Finally, we discuss the degree to which our results may be generalized.

Journal of Computational Neuroscience 40, 27-50 (2016). Journal linkWe examine the emergence of collective dynamical structures and complexity in a network of interacting populations of neuronal oscillators. Each population consists of a heterogeneous collection of globally-coupled theta neurons, which are a canonical representation of Type-1 neurons. For simplicity, the populations are arranged in a fully autonomous driver-response configuration, and we obtain a full description of the asymptotic macroscopic dynamics of this network. We find that the collective macroscopic behavior of the response population can exhibit equilibrium and limit cycle states, multistability, quasiperiodicity, and chaos, and we obtain detailed bifurcation diagrams that clarify the transitions between these macrostates. Furthermore, we show that despite the complexity that emerges, it is possible to understand the complicated dynamical structure of this system by building on the understanding of the collective behavior of a single population of theta neurons. This work is a first step in the construction of a mathematically-tractable network-of-networks representation of neuronal network dynamics.

Frontiers in Computational Neuroscience 8, 145 (2014). Journal link (open access)—This paper was included in Mattei, T. A., ed. (2016).

Under certain conditions, the collective behavior of a large globally-coupled heterogeneous network of coupled oscillators, as quantified by the macroscopic mean field or order parameter, can exhibit low-dimensional chaotic behavior. Recent advances describe how a small set of reduced ordinary differential equations can be derived that captures this mean field behavior. Here, we show that chaos control algorithms designed using the reduced equations can be successfully applied to imperfect realizations of the full network. To systematically study the effectiveness of this technique, we measure the quality of control as we relax conditions that are required for the strict accuracy of the reduced equations, and hence, the controller. Although the effects are network-dependent, we show that the method is effective for surprisingly small networks, for modest departures from global coupling, and even with mild inaccuracy in the estimate of network heterogeneity.

Chaos 24, 023127 (2014). Journal link PDF.Using recently developed analytical techniques, we study the macroscopic dynamics of a large heterogeneous network of theta neurons in which the neurons' excitability parameter varies in time. We demonstrate that such periodic variation can lead to the emergence of macroscopic chaos, multistability, and final-state uncertainty in the collective behavior of the network. Finite-size network effects and rudimentary control via an accessible macroscopic network parameter is also investigated.

Physica D 267, 16-26 (2014). Journal link.We investigate Inverse Stochastic Resonance (ISR), a recently-reported phenomenon in which the spiking activity of a Hodgkin-Huxley model neuron subject to external noise exhibits a pronounced minimum as the noise intensity increases. We clarify the mechanism that underlies ISR and show that its most surprising features are a consequence of the dynamical structure of the model. Furthermore, we show that the ISR effect depends strongly on the procedures used to measure it. Our results are important for the experimentalist who seeks to observe the ISR phenomenon.

Physical Review E 88, 042712 (2013). A PDF is available. Journal link (open access).We design and analyze the dynamics of a large network of theta neurons, which are idealized Type-I neurons. The network is heterogeneous in that it includes both inherently spiking and excitable neurons. The coupling is global, via pulse-like synapses of adjustable sharpness. Using recently-developed analytical methods, we identify all possible asymptotic states that can be exhibited by a mean-field variable that captures the network's macroscopic state. These consist of two equilibrium states that reflect partial synchronization in the network, and a limit cycle state in which the degree of network synchronization oscillates in time. Our approach also permits a complete bifurcation analysis, which we carry out with respect to parameters that capture the degree of excitability of the neurons, the heterogeneity in the population, and the coupling strength (which can be excitatory or inhibitory). We find that the network typically tends towards the two macroscopic equilibrium states when the neuron's intrinsic dynamics and the network interactions reinforce one another. In contrast, the limit cycle state, bifurcations, and multistability tend to occur when there is competition between these network features. Finally, we show that our results are exhibited by finite network realizations of reasonable size.

Neural Computation 25, 3207-3234 (2013). Journal link Author-generated PDF AnimationsWe investigate the effects of adding periodic stimulation to a generic, conductance-based neuron model that includes ion concentration dynamics of sodium and potassium. Under conditions of high extracellular potassium, the model exhibits repeating, spontaneous, seizure-like bursting events associated with slow modulation of the ion concentrations local to the neuron. We show that for a range of parameter values, depolarizing and hyperpolarizing periodic stimulation pulses (including frequencies lower than 4 Hz) can stop the spontaneous bursting by interacting with the ion concentration dynamics. Stimulation can also control the magnitude of evoked responses to modeled physiological inputs. We develop an understanding of the nonlinear dynamics of this system by a dimension reduction procedure that identifies effective nullclines in the ion concentration parameter space. Our results suggest that the manipulation of ion concentration dynamics via external or endogenous stimulation may play an important role in neuronal excitability, seizure dynamics, and control.

PLoS ONE 8(9): e73820 (2013). Journal link (open access).Cortical oscillations arise during behavioral and mental tasks, and all temporal oscillations have particular spatial patterns. Studying the mechanisms that generate and modulate the spatiotemporal characteristics of oscillations is important for understanding neural information processing and the signs and symptoms of dynamical diseases of the brain. Nevertheless, it remains unclear how GABAergic inhibition modulates these oscillation dynamics. Using voltage sensitive dye imaging, pharmacological methods, and tangentially cut occipital neocortical brain slices (including layers 3-5) of Sprague-Dawley rat, we found that GABAa disinhibition with bicuculline can progressively simplify oscillation dynamics in the presence of carbachol in a concentration-dependent way. Additionally, GABAb disinhibition can further simplify oscillation dynamics after GABAa receptors are blocked. Both GABAa and GABAb disinhibition increase the synchronization of the neural network. By using a combination of GABAa and GABAb antagonists alone, theta frequency (5-15Hz) oscillations are reliably generated. These theta oscillations have basic spatiotemporal patterns similar to those generated by carbachol/bicuculline. These results are illustrative of how GABAergic inhibition increases the complexity of patterns of activity and contributes to the regulation of cortex.

European Journal of Neuroscience 36, 2201-2212 (2012). A PDF is available. Journal linkWe consider an infinite network of globally-coupled phase oscillators in which the natural frequencies of the oscillators are drawn from a symmetric bimodal distribution. We demonstrate that macroscopic chaos can occur in this system when the coupling strength varies periodically in time. We identify period-doubling cascades to chaos, attractor crises, and horseshoe dynamics for the macroscopic mean field. Based on recent work that clarified the bifurcation structure of the static bimodal Kuramoto system, we qualitatively describe the mechanism for the generation of such complicated behavior in the time varying case.

This paper appears in Chaos 21, 21, 033127 (2011). A PDF is available. Journal linkWe describe a simple conductance-based model neuron that includes intra- and extra-cellular ion concentration dynamics and show that this model exhibits periodic bursting. The bursting arises as the fast spiking behavior of the neuron is modulated by the slow oscillatory behavior in the ion concentration variables, and vice versa. By separating these time scales and studying the bifurcation structure of the neuron, we catalog several qualitatively different bursting profiles that are strikingly similar to those seen in experimental preparations. Our work suggests that ion concentration dynamics may play an important role in modulating neuronal excitability in real biological systems.

This paper appears in the Journal of Biological Physics 37, 361-373 (2011). An author-generated version is available. Journal linkMammalian prenatal neocortical development is dominated by the synchronized formation of the laminae and migration of neurons. Postnatal development likewise contains "sensitive periods" during which functions such as ocular dominance emerge. Here we introduce a novel neuroinformatics approach to identify and study these periods of active development. Although many aspects of the approach can be used in other studies, some specific techniques were chosen because of a legacy dataset of human histological data (Conel in "The postnatal development of the human cerebral cortex", vol 1-8. Harvard University Press, Cambridge, 1939-1967). Our method calculates normalized change vectors from the raw histological data, and then employs k-means cluster analysis of the change vectors to explore the population dynamics of neurons from 37 neocortical areas across eight postnatal developmental stages from birth to 72 months in 54 subjects. We show that the cortical "address" (Brodmann area/ sub-area and layer) provides the necessary resolution to segregate neuron population changes into seven correlated "k-clusters" in k-means cluster analysis. The members in each k-cluster share a single change interval where the relative share of the cortex by the members undergoes its maximum change. The maximum change occurs in a different change interval for each k-cluster. Each k-cluster has at least one totally connected maximal "clique" which appears to correspond to cortical function.

This paper appears in Cognitive Neurodynamics 4, 151-163 (2010). Journal linkWe analyze a large system of globally coupled phase oscillators whose natural frequencies are bimodally distributed. The dynamics of this system has been the subject of long-standing interest. In 1984 Kuramoto proposed several conjectures about its behavior; ten years later, Crawford obtained the first analytical results by means of a local center manifold calculation. Nevertheless, many questions have remained open, especially about the possibility of global bifurcations. Here we derive the system's complete stability diagram for the special case where the bimodal distribution consists of two equally weighted Lorentzians. Using an ansatz recently discovered by Ott and Antonsen, we show that in this case the infinite-dimensional problem reduces exactly to a flow in four dimensions. Depending on the parameters and initial conditions, the long-term dynamics evolves to one of three states: incoherence, where all the oscillators are desynchronized; partial synchrony, where a macroscopic group of phase-locked oscillators coexists with a sea of desynchronized ones; and a standing wave state, where two counter-rotating groups of phase-locked oscillators emerge. Analytical results are presented for the bifurcation boundaries between these states. Similar results are also obtained for the case in which the bimodal distribution is given by the sum of two Gaussians.

This paper appears in Physical Review E, Vol. 79, 026204 (2009).A PDF file of this paper is available.

In these companion papers, we study how the interrelated dynamics of sodium and potassium affect the excitability of neurons, the occurrence of seizures, and the stability of persistent states of activity. We seek to study these dynamics with respect to the following compartments: neurons, glia, and extracellular space. We are particularly interested in the slower time-scale dynamics that determine overall excitability, and set the stage for transient episodes of persistent oscillations, working memory, or seizures. In this second of two companion papers, we present an ionic current network model composed of populations of Hodgkin–Huxley type excitatory and inhibitory neurons embedded within extracellular space and glia, in order to investigate the role of micro-environmental ionic dynamics on the stability of persistent activity. We show that these networks reproduce seizure-like activity if glial cells fail to maintain the proper micro-environmental conditions surrounding neurons, and produce several experimentally testable predictions. Our work suggests that the stability of persistent states to perturbation is set by glial activity, and that how the response to such perturbations decays or grows may be a critical factor in a variety of disparate transient phenomena such as working memory, burst firing in neonatal brain or spinal cord, up states, seizures, and cortical oscillations.

This paper appears in the Journal of Computational Neuroscience, Vol. 26, 171-183 (2009). Journal linkIn these companion papers, we study how the interrelated dynamics of sodium and potassium affect the excitability of neurons, the occurrence of seizures, and the stability of persistent states of activity. In this first paper, we construct a mathematical model consisting of a single conductance-based neuron together with intra- and extracellular ion concentration dynamics. We formulate a reduction of this model that permits a detailed bifurcation analysis, and show that the reduced model is a reasonable approximation of the full model. We find that competition between intrinsic neuronal currents, sodium-potassium pumps, glia, and diffusion can produce very slow and large-amplitude oscillations in ion concentrations similar to what is seen physiologically in seizures. Using the reduced model, we identify the dynamical mechanisms that give rise to these phenomena. These models reveal several experimentally testable predictions. Our work emphasizes the critical role of ion concentration homeostasis in the proper functioning of neurons, and points to important fundamental processes that may underlie pathological states such as epilepsy.

In many networks of interest (including technological, biological, and social networks), the connectivity between the interacting elements is not static, but changes in time. Furthermore, the elements themselves are often not identical, but rather display a variety of behaviors, and may come in different classes. Here, we investigate the dynamics of such systems. Specifically, we study a network of two large interacting heterogeneous populations of limit-cycle oscillators whose connectivity switches between two fixed arrangements at a particular frequency. We show that for sufficiently high switching frequency, this system behaves as if the connectivity were static and equal to the time-average of the switching connectivity. We also examine the mechanisms by which this fast-switching limit is approached in several non-intuitive cases. The results illuminate novel mechanisms by which synchronization can arise or be thwarted in large populations of coupled oscillators with non-static coupling.

This paper appears in Chaos 18, 037114 (2008).A PDF file of this paper is available.

The onset of synchronization in networks of networks is investigated. Specifically, we consider networks of interacting phase oscillators in which the set of oscillators is composed of several distinct populations. The oscillators in a given population are heterogeneous in that their natural frequencies are drawn from a given distribution, and each population has its own such distribution. The coupling among the oscillators is global, however, we permit the coupling strengths between the members of different populations to be separately specified. We determine the critical condition for the onset of coherent collective behavior, and develop the illustrative case in which the oscillator frequencies are drawn from a set of (possibly different) Cauchy-Lorentz distributions. One motivation is drawn from neurobiology, in which the collective dynamics of several interacting populations of oscillators (such as excitatory and inhibitory neurons and glia) are of interest.

This paper appears in Physical Review E, Vol. 77, 036107 (2008).A PDF file of this paper is available.

Excitatory and inhibitory (EI) interactions shape network activity. However, little is known about the EI interactions in pathological conditions such as epilepsy. To investigate EI interactions during seizure-like events (SLEs), we performed simultaneous dual and triple whole cell and extracellular recordings in pyramidal cells and oriens interneurons in rat hippocampal CA1. We describe a novel pattern of interleaving EI activity during spontaneous in vitro SLEs generated by the potassium channel blocker 4-aminopyridine in the presence of decreased magnesium. Interneuron activity was increased during interictal periods. During ictal discharges, interneurons entered into long-lasting depolarization block (DB) with suppression of spike generation; simultaneously, pyramidal cells produced spike trains with increased frequency (6-14 Hz) and correlation. After this period of runaway excitation, interneuron postictal spiking resumed and pyramidal cells became progressively quiescent. We performed correlation measures of cell-pair interactions using either the spikes alone or the subthreshold postsynaptic interspike signals. EE spike correlation was notably increased during interneuron DB, whereas subthreshold EE correlation decreased. EI spike correlations increased at the end of SLEs, whereas II subthreshold correlations increased during DB. Our findings underscore the importance of complex cell-type-specific neuronal interactions in the formation of seizure patterns.

This paper appears in the Journal of Neurophysiology, Vol. 95, pp. 3948-3954 (2006). Journal linkModels with a time delay often occur, since there is a naturally occurring delay in the transmission of information. A model with a delay can be noninvertible, which in turn leads to qualitative differences between the dynamical properties of a delay equation and the familiar case of an ordinary differential equation. We give specifc conditions for the existence of noninvertible solutions in delay equations, and describe the consequences of noninvertibility.

This paper appears in Discrete and Continuous Dynamical Systems, Supplementary Volume, pp. 768-777 (August 2005). Journal link (open access)We examine the effects of applied electric fields on neuronal synchronization. Twocompartment model neurons were synaptically coupled and embedded within a resistive array, thus allowing the neurons to interact both chemically and electrically. In addition, an external electric field was imposed on the array. The effects of this field were found to be nontrivial, giving rise to domains of synchrony and asynchrony as a function of the heterogeneity among the neurons. A simple phase oscillator reduction was successful in qualitatively reproducing these domains. The findings form several readily testable experimental predictions, and the model can be extended to a larger scale in which the effects of electric fields on seizure activity may be simulated.

This paper appears in the Journal of Computational Neuroscience, Vol. 19 #1, pp. 53-70 (2005). Journal linkPeriodicity is ubiquitous in nature. In this work, we analyze the dynamical reasons for which periodic windows that appear in parameter space diagrams have different shapes and structures. For that, we make use of a dynamical quantity called spine -- the skeleton of the window -- in order to explain a conjecture that describes the presence of periodic windows in the parameter space of high-dimensional chaotic systems.

This paper appears in the International Journal of Bifurcation and Chaos, Vol 13 #9, p. 2681-2688 (2003). Journal link

Motivated by the observation that applied electric fields modulate hippocampal seizures, and that seizures may be asynchronous, we modeled synaptically-coupled 2-compartment hippocampal pyramidal neurons embedded within an electrically resistive lattice in order to examine network synchronization properties under the influence of externally applied electric fields. Excitatory electric fields were shown to synchronize or desynchronize the network depending on the natural frequency mismatch between the neurons. Such frequency mismatch was found to decrease as a function of increasing electric field amplitude. These findings provide testable hypotheses for future seizure control experiments.

This paper appears in Neurocomputing 52-54, pp. 169-175 (2003). Journal link

Chaos synchronization in coupled systems is often characterized by a map $\phi$ between the states of the components. In noninvertible systems, or in systems without inherent symmetries, the synchronization set -- by which we mean graph($\phi$) -- can be extremely complicated. We identify, describe, and give examples of several different complications that can arise, and we link each to inherent properties of the underlying dynamics. We also discuss how these features can be quantified, and how they interfere with standard methods for detecting synchronization in measured data.

This paper appears in Chaos 13 #1, pp. 151-164 (2003).

A PDF file of this paper is available.

A general stability analysis is presented for the determination of the transition from incoherent to coherent behavior in an ensemble of globally coupled, heterogeneous, continuous-time dynamical systems. The formalism allows for the simultaneous presence of ensemble members exhibiting chaotic and periodic behavior, and, in a special case, yields the Kuramoto model for globally coupled periodic oscillators described by a phase. Numerical experiments using different types of ensembles of Lorenz equations with a distribution of parameters are presented.

This paper appears in Physica D Vol. 173, pp. 29-51 (2002). Journal link

Chaos synchronization is often characterized by the existence of a continuous function between the states of the components. However, in coupled systems without inherent symmetries, the synchronization set can be extremely complicated. We describe and illustrate three typical complications that can arise, and we discuss how existing methods for detecting synchronization will be hampered by the presence of these features.

This paper appears in Physical Review E, Vol. 65, 046225 (2002).

A PDF file of this paper is available.

The synchronization of chaotic systems has received a great deal of attention. However, most of the literature has focused on systems that possess invariant manifolds that persist as the coupling is varied. In this paper, we describe the process whereby synchronization is lost in systems of non-identical coupled chaotic oscillators without special symmetries. We qualitatively and quantitatively analyze such systems in terms of the evolution of the unstable periodic orbit structure. Our results are illustrated with data from physical experiments.

This paper appears in the International Journal of Bifurcation and Chaos, Vol 11 #10, p. 2705-2713 (2001). Journal link

A chaotic attractor containing unstable periodic orbits with different numbers of unstable directions is said to exhibit unstable dimension variability (UDV). We present a general mechanism for the progressive development of UDV in uni- and bidirectionally coupled systems of chaotic elements. Our results are applicable to systems of dissimilar elements without invariant manifolds. We also quantify the severity of UDV to identify coupling ranges where the shadowability and modelability of such systems are significantly compromised.

This paper appears in
Physical Review Letters, **85**
#12, p.
2490-2493 (18 September 2000).

A PDF file of this paper is available.

Emergent Sets in Coupled Chaotic Systems

We consider the evolution of the unstable periodic orbit structure of coupled chaotic systems. This involves the creation of a complicated set outside of the synchronization manifold (the emergent set). We quantitatively identify a critical transition point in its development (the decoherence transition). For asymmetric systems we also describe a migration of unstable periodic orbits that is of central importance in understanding these systems. Our framework provides an experimentally measurable transition, even in situations where previously described bifurcation structures are inapplicable.

This paper appears in
Physical Review Letters **84**
#8, p.
1689-1692 (21 February 2000).

A PDF file of this paper is available.

Computing D_0 from Average Expansion Rates

We propose an iterative scheme for calculating the box-counting (capacity) dimension of a chaotic attractor in terms of its average expansion rates. Similar to the Kaplan-Yorke conjecture for the information dimension, this scheme provides a connection between a geometric property of a strange set and its underlying dynamical properties. Our conjecture is demonstrated analytically with an exactly solvable two dimensional hyperbolic map and numerically with a more complicated higher dimensional non-hyperbolic map.

This paper appears in
Physical Review E **60**
#1, p. 378-385
(July 1999).

A PDF file of this paper is available.

The Structure of Parameter Space

Regions in the parameter space of chaotic systems that correspond to stable behavior are often referred to as windows. In this Letter, we elucidate the occurrence of such regions in higher dimensional chaotic systems. We describe the fundamental structure of these windows, and also indicate under what circumstances one can expect to find them. These results are applicable to systems that exhibit several positive Lyapunov exponents, and are of importance to both the theoretical and the experimental understanding of dynamical systems.

This paper appears in
Physical Review Letters **78**
#24, p.
4561-4564 (16 June 1997).

A PDF file of this paper is available.

Multiparameter Control of Chaos

Controlling chaos by using more than one available control parameter is presented as an experimentally feasable way to reduce the transient times that precede stabilization and improve performance in the presence of noise. We demonstrate these advantages by applying our method to a numerical model of a physical system.

This paper appears in
Physical Review E **52**
#4, p.
3553-3557 (October 1995).

We develop an efficient targeting technique and demonstrate that when used with an unstable periodic orbit stabilization method, fast and efficient switching between controlled periodic orbits is possible. This technique is particularly relevant to cases of higher attractor dimension. We present a numerical example and report an improvement of up to 4 orders of magnitude in the switching time over the case with no targeting.

This paper appears in
Physical Review E **51**
#5, p.
4169-4172 (May 1995).