Geometric analysis of population rhythms in synaptically coupled neuronal networks

Nonlinear mechanism for the enhanced bursting activities induced by fast inhibitory autapse and reduced activities by fast excitatory autapse

The paradoxical phenomena that excitatory modulation does not enhance but reduces or inhibitory modulation not suppresses but promotes neural firing activities have attracted increasing attention. In the present study, paradoxical phenomena induced by both fast excitatory and inhibitory autapses in a "Fold/Big Homoclinic" bursting are simulated, and the corresponding nonlinear and biophysical mechanisms are presented. Firstly, the enhanced conductance of excitatory autapse induces the number of spikes per burst and firing rate reduced, while the enhanced inhibitory autapse cause both indicators increased. Secondly, with fast-slow variable dissection, the burst of bursting is identified to locate between a fold bifurcation and a big saddle-homoclinic orbit bifurcation of the fast subsystem. Enhanced excitatory or inhibitory autapses cannot induce changes of both bifurcation points, i.e., burst width. However, width of slow variable between two successive spikes within a burst becomes wider for the excitatory autapse and narrower for the inhibitory autapse, resulting in the less and more spikes per burst, respectively. Last, the autaptic current of fast autapse mainly plays a role during the peak of action potential, differing from the slow autaptic current with exponential decay, which can play roles following the peak of action potential. The fast excitatory autaptic current enhances the amplitude of the action potential and reduces the repolarization of the action potential to lengthen the interspike interval (ISI) of the spiking of the fast subsystem, resulting in the wide width of slow variable between successive spikes. The fast inhibitory autaptic current reduces the amplitude of action potential and ISI of spiking, resulting in narrow width of slow variable. The novel example of the paradoxical responses for both fast modulations and nonlinear mechanism extend the contents of neurodynamics, which presents potential functions of the fast autapse.

Hierarchical processing underpins competition in tactile perceptual bistability

Ambiguous sensory information can lead to spontaneous alternations between perceptual states, recently shown to extend to tactile perception. The authors recently proposed a simplified form of tactile rivalry which evokes two competing percepts for a fixed difference in input amplitudes across antiphase, pulsatile stimulation of the left and right fingers. This study addresses the need for a tactile rivalry model that captures the dynamics of perceptual alternations and that incorporates the structure of the somatosensory system. The model features hierarchical processing with two stages. The first and the second stages of model could be located at the secondary somatosensory cortex (area S2), or in higher areas driven by S2. The model captures dynamical features specific to the tactile rivalry percepts and produces general characteristics of perceptual rivalry: input strength dependence of dominance times (Levelt's proposition II), short-tailed skewness of dominance time distributions and the ratio of distribution moments. The presented modelling work leads to experimentally testable predictions. The same hierarchical model could generalise to account for percept formation, competition and alternations for bistable stimuli that involve pulsatile inputs from the visual and auditory domains.

Auditory streaming emerges from fast excitation and slow delayed inhibition

In the auditory streaming paradigm, alternating sequences of pure tones can be perceived as a single galloping rhythm (integration) or as two sequences with separated low and high tones (segregation). Although studied for decades, the neural mechanisms underlining this perceptual grouping of sound remains a mystery. With the aim of identifying a plausible minimal neural circuit that captures this phenomenon, we propose a firing rate model with two periodically forced neural populations coupled by fast direct excitation and slow delayed inhibition. By analyzing the model in a non-smooth, slow-fast regime we analytically prove the existence of a rich repertoire of dynamical states and of their parameter dependent transitions. We impose plausible parameter restrictions and link all states with perceptual interpretations. Regions of stimulus parameters occupied by states linked with each percept match those found in behavioural experiments. Our model suggests that slow inhibition masks the perception of subsequent tones during segregation (forward masking), whereas fast excitation enables integration for large pitch differences between the two tones.

Dynamics and bifurcations in multistable 3-cell neural networks

We disclose the generality of the intrinsic mechanisms underlying multistability in reciprocally inhibitory 3-cell circuits composed of simplified, low-dimensional models of oscillatory neurons, as opposed to those of a detailed Hodgkin-Huxley type [Wojcik et al., PLoS One 9, e92918 (2014)]. The computational reduction to return maps for the phase-lags between neurons reveals a rich multiplicity of rhythmic patterns in such circuits. We perform a detailed bifurcation analysis to show how such rhythms can emerge, disappear, and gain or lose stability, as the parameters of the individual cells and the synapses are varied.

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 synchronized behaviour. We consider a network of excitable, relaxation oscillator neurons where two distinct populations, one excitatory and one inhibitory, are coupled with time-delayed synapses. The excitatory population is uncoupled, while the inhibitory population is tightly coupled without time delay. A geometric singular perturbation analysis yields existence and stability conditions for periodic solutions where the excitatory cells are synchronized and different phase relationships between the excitatory and inhibitory populations can occur, along with formulae for the periods of such solutions. In particular, we show that if there are no delays in the coupling, oscillations where the excitatory population is synchronized cannot occur. Numerical simulations are conducted to supplement and validate the analytical results. The analysis helps to explain how coupling delays in either excitatory or inhibitory synapses contribute to producing synchronized rhythms. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

When two wrongs make a right: synchronized neuronal bursting from combined electrical and inhibitory coupling

Synchronized cortical activities in the central nervous systems of mammals are crucial for sensory perception, coordination and locomotory function. The neuronal mechanisms that generate synchronous synaptic inputs in the neocortex are far from being fully understood. In this paper, we study the emergence of synchronization in networks of bursting neurons as a highly non-trivial, combined effect of electrical and inhibitory connections. We report a counterintuitive find that combined electrical and inhibitory coupling can synergistically induce robust synchronization in a range of parameters where electrical coupling alone promotes anti-phase spiking and inhibition induces anti-phase bursting. We reveal the underlying mechanism, which uses a balance between hidden properties of electrical and inhibitory coupling to act together to synchronize neuronal bursting. We show that this balance is controlled by the duty cycle of the self-coupled system which governs the synchronized bursting rhythm.This article is part of the themed issue 'Mathematical methods in medicine: neuroscience, cardiology and pathology'.

Dynamics of Time Delay-Induced Multiple Synchronous Behaviors in Inhibitory Coupled Neurons

The inhibitory synapse can induce synchronous behaviors different from the anti-phase synchronous behaviors, which have been reported in recent studies. In the present paper, synchronous behaviors are investigated in the motif model composed of reciprocal inhibitory coupled neurons with endogenous bursting and time delay. When coupling strength is weak, synchronous behavior appears at a single interval of time delay within a bursting period. When coupling strength is strong, multiple synchronous behaviors appear at different intervals of time delay within a bursting period. The different bursting patterns of synchronous behaviors, and time delays and coupling strengths that can induce the synchronous bursting patterns can be well interpreted by the dynamics of the endogenous bursting pattern of isolated neuron, which is acquired by the fast-slow dissection method, combined with the inhibitory coupling current. For an isolated neuron, when a negative impulsive current with suitable strength is applied at different phases of the bursting, multiple different bursting patterns can be induced. For a neuron in the motif, the inhibitory coupling current, of which the application time and strength is modulated by time delay and coupling strength, can cause single or multiple synchronous firing patterns like the negative impulsive current when time delay and coupling strength is suitable. The difference compared to the previously reported multiple synchronous behaviors that appear at time delays wider than a period of the endogenous firing is discussed. The results present novel examples of synchronous behaviors in the neuronal network with inhibitory synapses and provide a reasonable explanation.

Synergistic effect of repulsive inhibition in synchronization of excitatory networks

We show that the addition of pairwise repulsive inhibition to excitatory networks of bursting neurons induces synchrony, in contrast to one's expectations. Through stability analysis, we reveal the mechanism underlying this purely synergistic phenomenon and demonstrate that it originates from the transition between different types of bursting, caused by excitatory-inhibitory synaptic coupling. This effect is generic and observed in different models of bursting neurons and fast synaptic interactions. We also find a universal scaling law for the synchronization stability condition for large networks in terms of the number of excitatory and inhibitory inputs each neuron receives, regardless of the network size and topology. This general law is in sharp contrast with linearly coupled networks with positive (attractive) and negative (repulsive) coupling where the placement and structure of negative connections heavily affect synchronization.

Frequency Preference Response to Oscillatory Inputs in Two-dimensional Neural Models: A Geometric Approach to Subthreshold Amplitude and Phase Resonance

We investigate the dynamic mechanisms of generation of subthreshold and phase resonance in two-dimensional linear and linearized biophysical (conductance-based) models, and we extend our analysis to account for the effect of simple, but not necessarily weak, types of nonlinearities. Subthreshold resonance refers to the ability of neurons to exhibit a peak in their voltage amplitude response to oscillatory input currents at a preferred non-zero (resonant) frequency. Phase-resonance refers to the ability of neurons to exhibit a zero-phase (or zero-phase-shift) response to oscillatory input currents at a non-zero (phase-resonant) frequency. We adapt the classical phase-plane analysis approach to account for the dynamic effects of oscillatory inputs and develop a tool, the envelope-plane diagrams, that captures the role that conductances and time scales play in amplifying the voltage response at the resonant frequency band as compared to smaller and larger frequencies. We use envelope-plane diagrams in our analysis. We explain why the resonance phenomena do not necessarily arise from the presence of imaginary eigenvalues at rest, but rather they emerge from the interplay of the intrinsic and input time scales. We further explain why an increase in the time-scale separation causes an amplification of the voltage response in addition to shifting the resonant and phase-resonant frequencies. This is of fundamental importance for neural models since neurons typically exhibit a strong separation of time scales. We extend this approach to explain the effects of nonlinearities on both resonance and phase-resonance. We demonstrate that nonlinearities in the voltage equation cause amplifications of the voltage response and shifts in the resonant and phase-resonant frequencies that are not predicted by the corresponding linearized model. The differences between the nonlinear response and the linear prediction increase with increasing levels of the time scale separation between the voltage and the gating variable, and they almost disappear when both equations evolve at comparable rates. In contrast, voltage responses are almost insensitive to nonlinearities located in the gating variable equation. The method we develop provides a framework for the investigation of the preferred frequency responses in three-dimensional and nonlinear neuronal models as well as simple models of coupled neurons.

Key bifurcations of bursting polyrhythms in 3-cell central pattern generators

We identify and describe the key qualitative rhythmic states in various 3-cell network motifs of a multifunctional central pattern generator (CPG). Such CPGs are neural microcircuits of cells whose synergetic interactions produce multiple states with distinct phase-locked patterns of bursting activity. To study biologically plausible CPG models, we develop a suite of computational tools that reduce the problem of stability and existence of rhythmic patterns in networks to the bifurcation analysis of fixed points and invariant curves of a Poincaré return maps for phase lags between cells. We explore different functional possibilities for motifs involving symmetry breaking and heterogeneity. This is achieved by varying coupling properties of the synapses between the cells and studying the qualitative changes in the structure of the corresponding return maps. Our findings provide a systematic basis for understanding plausible biophysical mechanisms for the regulation of rhythmic patterns generated by various CPGs in the context of motor control such as gait-switching in locomotion. Our analysis does not require knowledge of the equations modeling the system and provides a powerful qualitative approach to studying detailed models of rhythmic behavior. Thus, our approach is applicable to a wide range of biological phenomena beyond motor control.

Swing, release, and escape mechanisms contribute to the generation of phase-locked cluster patterns in a globally coupled FitzHugh-Nagumo model

We investigate the mechanism of generation of phase-locked cluster patterns in a globally coupled FitzhHugh-Nagumo model where the fast variable (activator) receives global feedback from the slow variable (inhibitor). We identify three qualitatively different mechanisms (swing-and-release, hold-and-release, and escape-and-release) that contribute to the generation of these patterns. We describe these mechanisms and use this framework to explain under what circumstances two initially out-of-phase oscillatory clusters reach steady phase-locked and in-phase synchronized solutions, and how the phase difference between these steady state cluster patterns depends on the clusters relative size, the global coupling intensity, and other model parameters.

Spikes matter for phase-locked bursting in inhibitory neurons

We show that inhibitory networks composed of two endogenously bursting neurons can robustly display several coexistent phase-locked states in addition to stable antiphase and in-phase bursting. This work complements and enhances our recent result [Jalil, Belykh, and Shilnikov, Phys. Rev. E 81, 045201(R) (2010)] that fast reciprocal inhibition can synchronize bursting neurons due to spike interactions. We reveal the role of spikes in generating multiple phase-locked states and demonstrate that this multistability is generic by analyzing diverse models of bursting networks with various fast inhibitory synapses; the individual cell models include the reduced leech heart interneuron, the Sherman model for pancreatic beta cells, and the Purkinje neuron model.

Shaping bursting by electrical coupling and noise

Gap-junctional coupling is an important way of communication between neurons and other excitable cells. Strong electrical coupling synchronizes activity across cell ensembles. Surprisingly, in the presence of noise synchronous oscillations generated by an electrically coupled network may differ qualitatively from the oscillations produced by uncoupled individual cells forming the network. A prominent example of such behavior is the synchronized bursting in islets of Langerhans formed by pancreatic β-cells, which in isolation are known to exhibit irregular spiking (Sherman and Rinzel, Biophys J 54:411-425, 1988; Sherman and Rinzel, Biophys J 59:547-559, 1991). At the heart of this intriguing phenomenon lies denoising, a remarkable ability of electrical coupling to diminish the effects of noise acting on individual cells. In this paper, building on an earlier analysis of denoising in networks of integrate-and-fire neurons (Medvedev, Neural Comput 21 (11):3057-3078, 2009) and our recent study of spontaneous activity in a closely related model of the Locus Coeruleus network (Medvedev and Zhuravytska, The geometry of spontaneous spiking in neuronal networks, submitted, 2012), we derive quantitative estimates characterizing denoising in electrically coupled networks of conductance-based models of square wave bursting cells. Our analysis reveals the interplay of the intrinsic properties of the individual cells and network topology and their respective contributions to this important effect. In particular, we show that networks on graphs with large algebraic connectivity (Fiedler, Czech Math J 23(98):298-305, 1973) or small total effective resistance (Bollobas, Modern graph theory, Graduate Texts in Mathematics, vol. 184, Springer, New York, 1998) are better equipped for implementing denoising. As a by-product of the analysis of denoising, we analytically estimate the rate with which trajectories converge to the synchronization subspace and the stability of the latter to random perturbations. These estimates reveal the role of the network topology in synchronization. The analysis is complemented by numerical simulations of electrically coupled conductance-based networks. Taken together, these results explain the mechanisms underlying synchronization and denoising in an important class of biological models.

Model-based analysis and control of a network of basal ganglia spiking neurons in the normal and parkinsonian states

Controlling the spatiotemporal firing pattern of an intricately connected network of neurons through microstimulation is highly desirable in many applications. We investigated in this paper the feasibility of using a model-based approach to the analysis and control of a basal ganglia (BG) network model of Hodgkin-Huxley (HH) spiking neurons through microstimulation. Detailed analysis of this network model suggests that it can reproduce the experimentally observed characteristics of BG neurons under a normal and a pathological Parkinsonian state. A simplified neuronal firing rate model, identified from the detailed HH network model, is shown to capture the essential network dynamics. Mathematical analysis of the simplified model reveals the presence of a systematic relationship between the network's structure and its dynamic response to spatiotemporally patterned microstimulation. We show that both the network synaptic organization and the local mechanism of microstimulation can impose tight constraints on the possible spatiotemporal firing patterns that can be generated by the microstimulated network, which may hinder the effectiveness of microstimulation to achieve a desired objective under certain conditions. Finally, we demonstrate that the feedback control design aided by the mathematical analysis of the simplified model is indeed effective in driving the BG network in the normal and Parskinsonian states to follow a prescribed spatiotemporal firing pattern. We further show that the rhythmic/oscillatory patterns that characterize a dopamine-depleted BG network can be suppressed as a direct consequence of controlling the spatiotemporal pattern of a subpopulation of the output Globus Pallidus internalis (GPi) neurons in the network. This work may provide plausible explanations for the mechanisms underlying the therapeutic effects of deep brain stimulation (DBS) in Parkinson's disease and pave the way towards a model-based, network level analysis and closed-loop control and optimization of DBS parameters, among many other applications.

Fast reciprocal inhibition can synchronize bursting neurons

We study a pair of endogenously bursting neurons with fast nondelayed inhibitory connections. We show that fast reciprocal inhibition, known to facilitate antiphase bursting, can stably synchronize bursting neurons. This contrasts with the classical view that reciprocal inhibition has to be slow or time delayed to establish in-phase synchronization. Through stability analysis, we reveal the emergent mechanism of in-phase synchronization and discuss its implications for various types of bursting neurons and networks.

Polyrhythmic synchronization in bursting networking motifs

We study the emergence of polyrhythmic dynamics of motifs which are the building block for small inhibitory-excitatory networks, such as central pattern generators controlling various locomotive behaviors of animals. We discover that the pacemaker determining the specific rhythm of such a network composed of realistic Hodgkin-Huxley-type neurons is identified through the order parameter, which is the ratio of the neurons' burst durations or of duty cycles. We analyze different configurations of the motifs and describe the universal mechanisms for synergetics of the bursting patterns. We discuss also the multistability of inhibitory networks that results in polyrhythmicity of its emergent synchronous behaviors.

Loss of phase-locking in non-weakly coupled inhibitory networks of type-I model neurons

Synchronization of excitable cells coupled by reciprocal inhibition is a topic of significant interest due to the important role that inhibitory synaptic interaction plays in the generation and regulation of coherent rhythmic activity in a variety of neural systems. While recent work revealed the synchronizing influence of inhibitory coupling on the dynamics of many networks, it is known that strong coupling can destabilize phase-locked firing. Here we examine the loss of synchrony caused by an increase in inhibitory coupling in networks of type-I Morris-Lecar model oscillators, which is characterized by a period-doubling cascade and leads to mode-locked states with alternation in the firing order of the two cells, as reported recently by Maran and Canavier (J Comput Nerosci, 2008) for a network of Wang-Buzsáki model neurons. Although alternating-order firing has been previously reported as a near-synchronous state, we show that the stable phase difference between the spikes of the two Morris-Lecar cells can constitute as much as 70% of the unperturbed oscillation period. Further, we examine the generality of this phenomenon for a class of type-I oscillators that are close to their excitation thresholds, and provide an intuitive geometric description of such "leap-frog" dynamics. In the Morris-Lecar model network, the alternation in the firing order arises under the condition of fast closing of K( + ) channels at hyperpolarized potentials, which leads to slow dynamics of membrane potential upon synaptic inhibition, allowing the presynaptic cell to advance past the postsynaptic cell in each cycle of the oscillation. Further, we show that non-zero synaptic decay time is crucial for the existence of leap-frog firing in networks of phase oscillators. However, we demonstrate that leap-frog spiking can also be obtained in pulse-coupled inhibitory networks of one-dimensional oscillators with a multi-branched phase domain, for instance in a network of quadratic integrate-and-fire model cells. Finally, for the case of a homogeneous network, we establish quantitative conditions on the phase resetting properties of each cell necessary for stable alternating-order spiking, complementing the analysis of Goel and Ermentrout (Physica D 163:191-216, 2002) of the order-preserving phase transition map.

Modulation of cortical oscillatory activity during transcranial magnetic stimulation

Transcranial magnetic stimulation (TMS) can transiently modulate cortical excitability, with a net effect depending on the stimulation frequency (< or =1 Hz inhibition vs. > or =5 Hz facilitation, at least for the motor cortex). This possibility has generated interest in experiments aiming to improve deficits in clinical settings, as well as deficits in the cognitive domain. The aim of the present study was to investigate the on-line effects of low frequency (1 Hz) TMS on the EEG oscillatory activity in the healthy human brain, focusing particularly on the outcome of these modulatory effects in relation to the duration of the TMS stimulation. To this end, we used the event-related desynchronization/synchronization (ERD/ERS) approach to determine the patterns of oscillatory activity during two consecutive trains of sham and real TMS. Each train of stimulation was delivered to the left primary motor cortex (MI) of healthy subjects over a period of 10 min, while EEG rhythms were simultaneously recorded. Results indicated that TMS induced an increase in the power of brain rhythms that was related to the period of the stimulation, i.e. the synchronization of the alpha band increased with the duration of the stimulation, and this increase was inversely correlated with motor-evoked potentials (MEPs) amplitude. In conclusion, low frequency TMS over primary motor cortex induces a synchronization of the background oscillatory activity on the stimulated region. This induced modulation in brain oscillations seems to increase coherently with the duration of stimulation, suggesting that TMS effects may involve short-term modification of the neural circuitry sustaining MEPs characteristics.

Transitions between irregular and rhythmic firing patterns in excitatory-inhibitory neuronal networks

Changes in firing patterns are an important hallmark of the functional status of neuronal networks. We apply dynamical systems methods to understand transitions between irregular and rhythmic firing in an excitatory-inhibitory neuronal network model. Using the geometric theory of singular perturbations, we systematically reduce the full model to a simpler set of equations, one that can be studied analytically. The analytic tools are used to understand how an excitatory-inhibitory network with a fixed architecture can generate both activity patterns for possibly different values of the intrinsic and synaptic parameters. These results are applied to a recently developed model for the subthalamopallidal network of the basal ganglia. The results suggest that an increase in correlated activity, corresponding to a pathological state, may be due to an increased level of inhibition from the striatum to the inhibitory GPe cells along with an increased ability of the excitatory STN neurons to generate rebound bursts.

Capturing the bursting dynamics of a two-cell inhibitory network using a one-dimensional map

Out-of-phase bursting is a functionally important behavior displayed by central pattern generators and other neural circuits. Understanding this complex activity requires the knowledge of the interplay between the intrinsic cell properties and the properties of synaptic coupling between the cells. Here we describe a simple method that allows us to investigate the existence and stability of anti-phase bursting solutions in a network of two spiking neurons, each possessing a T-type calcium current and coupled by reciprocal inhibition. We derive a one-dimensional map which fully characterizes the genesis and regulation of anti-phase bursting arising from the interaction of the T-current properties with the properties of synaptic inhibition. This map is the burst length return map formed as the composition of two distinct one-dimensional maps that are each regulated by a different set of model parameters. Although each map is constructed using the properties of a single isolated model neuron, the composition of the two maps accurately captures the behavior of the full network. We analyze the parameter sensitivity of these maps to determine the influence of both the intrinsic cell properties and the synaptic properties on the burst length, and to find the conditions under which multistability of several bursting solutions is achieved. Although the derivation of the map relies on a number of simplifying assumptions, we discuss how the principle features of this dimensional reduction method could be extended to more realistic model networks.

Bursting induced by excitatory synaptic coupling in nonidentical conditional relaxation oscillators or square-wave bursters

This work explains a mechanism through which the introduction of excitatory synaptic coupling between two model cells, one of which is excitable and the other of which is tonically active when uncoupled, leads to bursting in the resulting two-cell network. This phenomenon can arise when the individual cells are conditional relaxation oscillators, in that they can be tuned to engage in relaxation oscillations, or when they are conditional square-wave bursters. The mechanism is illustrated with a model for conditional pacemaker neurons in the pre-Bötzinger complex as well as with a reduced form of this model. In the relaxation oscillator case, a periodic bursting solution is proved to exist in the singular limit, under a pair of general conditions. These conditions relate the durations of the silent and active phases of the bursting solution to the locations of certain structures in the phase plane, at appropriate synaptic input strengths. Further, additional conditions on the relative flow rates in the silent and active phases are proved to imply the uniqueness and asymptotic stability of the bursting solution.

Local network parameters can affect inter-network phase lags in central pattern generators

Weakly coupled phase oscillators and strongly coupled relaxation oscillators have different mechanisms for creating stable phase lags. Many oscillations in central pattern generators combine features of each type of coupling: local networks composed of strongly coupled relaxation oscillators are weakly coupled to similar local networks. This paper analyzes the phase lags produced by this combination of mechanisms and shows how the parameters of a local network, such as the decay time of inhibition, can affect the phase lags between the local networks. The analysis is motivated by the crayfish central pattern generator used for swimming, and uses techniques from geometrical singular perturbation theory.

Cluster formation for multi-strain infections with cross-immunity

Many infectious diseases exist in several pathogenic variants, or strains, which interact via cross-immunity. It is observed that strains tend to self-organise into groups, or clusters. The aim of this paper is to investigate cluster formation. Computations demonstrate that clustering is independent of the model used, and is an intrinsic feature of the strain system itself. We observe that an ordered strain system, if it is sufficiently complex, admits several cluster structures of different types. Appearance of a particular cluster structure depends on levels of cross-immunity and, in some cases, on initial conditions. Clusters, once formed, are stable, and behave remarkably regularly (in contrast to the generally chaotic behaviour of the strains themselves). In general, clustering is a type of self-organisation having many features in common with pattern formation.

Using heterogeneity to predict inhibitory network model characteristics

From modeling studies it has been known for >10 years that purely inhibitory networks can produce synchronous output given appropriate balances of intrinsic and synaptic parameters. Several experimental studies indicate that synchronous activity produced by inhibitory networks is critical to the production of population rhythms associated with various behavioral states. Heterogeneity of inputs to inhibitory networks strongly affect their ability to synchronize. In this paper, we explore how the amount of input heterogeneity to two-cell inhibitory networks affects their dynamics. Using numerical simulations and bifurcation analyses, we find that the ability of inhibitory networks to synchronize in the face of heterogeneity depends nonmonotonically on each of the synaptic time constant, synaptic conductance and external drive parameters. Because of this, an optimal set of parameters for a given cellular model with various biophysical characteristics can be determined. We suggest that this could be a helpful approach to use in determining the importance of different, underlying biophysical details. We further find that two-cell coherence properties are maintained in larger 10-cell networks. As such, we think that a strategy of "embedding" small network dynamics in larger networks is a useful way to understand the contribution of biophysically derived parameters to population dynamics in large networks.

Emergent properties of CNS neuronal networks as targets for pharmacology: application to anticonvulsant drug action

CNS drugs may act by modifying the emergent properties of complex CNS neuronal networks. Emergent properties are network characteristics that are not predictably based on properties of individual member neurons. Neuronal membership within networks is controlled by several mechanisms, including burst firing, gap junctions, endogenous and exogenous neuroactive substances, extracellular ions, temperature, interneuron activity, astrocytic integration and external stimuli. The effects of many CNS drugs in vivo may critically involve actions on specific brain loci, but this selectivity may be absent when the same neurons are isolated from the network in vitro where emergent properties are lost. Audiogenic seizures (AGS) qualify as an emergent CNS property, since in AGS the acoustic stimulus evokes a non-linear output (motor convulsion), but the identical stimulus evokes minimal behavioral changes normally. The hierarchical neuronal network, subserving AGS in rodents is initiated in inferior colliculus (IC) and progresses to deep layers of superior colliculus (DLSC), pontine reticular formation (PRF) and periaqueductal gray (PAG) in genetic and ethanol withdrawal-induced AGS. In blocking AGS, certain anticonvulsants reduce IC neuronal firing, while other agents act primarily on neurons in other AGS network sites. However, the NMDA receptor channel blocker, MK-801, does not depress neuronal firing in any network site despite potently blocking AGS. Recent findings indicate that MK-801 actually enhances firing in substantia nigra reticulata (SNR) neurons in vivo but not in vitro. Thus, the MK-801-induced firing increases in SNR neurons observed in vivo may involve an indirect effect via disinhibition, involving an action on the emergent properties of this seizure network.

High-frequency, depressing inhibition facilitates synchronization in globally inhibitory networks

Motivated by the study of sharp wave-associated ripples, high-frequency (approximately 200 Hz) extracellular field oscillations observed in the CA1 region of the rat hippocampus during slow-wave sleep and periods of behavioural immobility, we consider a single inhibitory neuron synapsing onto a network of uncoupled, excitatory neurons. The inhibitory synapse is depressing and has a small synaptic delay. Each excitatory cell provides instantaneous, positive feedback to the inhibitory cell. We show that the interneuron can rapidly synchronize the action potentials of the pyramidal cells if the frequency of inhibitory input is increased in a ramp-like manner as occurs during the ripple. We show that the basin of attraction of the synchronous solution is larger when the inhibition frequency is gradually increased as opposed to remaining constant.

Short duty cycle destabilizes a half-center oscillator, but gap junctions can restabilize the anti-phase pattern

Mutually inhibitory pacemaker neurons with duty cycle close to 50% operate as a half-center oscillator (anti-phase coordination, i.e., 180 degrees out of phase), even in the presence of weak to modest gap junctional coupling. For electrical coupling strength above a critical value synchronization occurs. But, as shown here with modeling studies, the effects of electrical coupling depend critically on a cell's duty cycle. Instead of oscillating either in-phase or anti-phase, model cells with short duty cycle express additional rhythmic patterns, and different transitions between them, depending on electrical coupling strength. For weak or no electrical coupling, cells do not oscillate in anti-phase but instead exhibit almost in-phase activity. Strengthening this weak coupling leads to stable anti-phase activity. With yet stronger electrical coupling stable inphase (synchrony) emerges but it coexists with the anti-phase pattern. Thus the network shows bistability for an intermediate range of coupling strength. For sufficiently strong electrical coupling synchrony is the network's only attracting rhythmic state. Our results, numerical and analytical (phase plane analysis), are based on a minimal but biophysically motivated pacemaker model for the slowly oscillating envelope of bursting neurons. However, illustrations for an Hodgkin-Huxley model suggest that some of our results for short duty cycle may extend to patterning of repetitive spikes. In particular, electrical coupling of intermediate strength may promote anti-phase activity and provide bistability of anti-phase and in-phase spiking.

Dendritic synchrony and transient dynamics in a coupled oscillator model of the dopaminergic neuron

Transient increases in spontaneous firing rate of mesencephalic dopaminergic neurons have been suggested to act as a reward prediction error signal. A mechanism previously proposed involves subthreshold calcium-dependent oscillations in all parts of the neuron. In that mechanism, the natural frequency of oscillation varies with diameter of cell processes, so there is a wide variation of natural frequencies on the cell, but strong voltage coupling enforces a single frequency of oscillation under resting conditions. In previous work, mathematical analysis of a simpler system of oscillators showed that the chain of oscillators could produce transient dynamics in which the frequency of the coupled system increased temporarily, as seen in a biophysical model of the dopaminergic neuron. The transient dynamics was shown to be consequence of a slow drift along an invariant subset of phase space, with rate of drift given by a Lyapunov function. In this paper, we show that the same mathematical structure exists for the full biophysical model, giving physiological meaning to the slow drift and the Lyapunov function, which is shown to describe differences in intracellular calcium concentration in different parts of the cell. The duration of transients was long, being comparable to the time constant of calcium disposition. These results indicate that brief changes in input to the dopaminergic neuron can produce long lasting firing rate transients whose form is determined by intrinsic cell properties.

Continuum of weakly coupled oscillatory McKean neurons

The McKean model of a neuron possesses a one-dimensional fast voltagelike variable and a slow recovery variable. A recent geometric analysis of the singularly perturbed system has allowed an explicit construction of its phase response curve [S. Coombes, Physica D 160, 173 (2001)]. Here we use tools from coupled oscillator theory to study weakly coupled networks of McKean neurons. Using numerical techniques, we show that the McKean system has traveling wave phase-locked solutions consistent with that of a network of more biophysically detailed Hodgkin-Huxley neurons.

The effects of subthreshold 1 Hz repetitive TMS on cortico-cortical and interhemispheric coherence

OBJECTIVES

Repetitive transcranial magnetic stimulation (rTMS) shows promise as a treatment for various movement and psychiatric disorders. Just how rTMS may have persistent effects on cortical function remains unclear. We hypothesised that it may act by modulating cortico-cortical and interhemispheric connectivity. To this end we assessed cortico-cortical and interhemispheric coherence before and after low frequency, subthreshold rTMS of the left motor cortex.

METHODS

Fifteen healthy subjects received one train (1Hz, 90% of active motor threshold, 1500 stimuli) of rTMS to the left motor hand area. Spectral power and coherence estimates were calculated between different electroencephalogram (EEG) signals at rest and while muscles of the distal upper limb were tonically contracted.

RESULTS

rTMS over the left motor hand area caused a significant increase in ipsilateral EEG-EEG coherence and in the interhemispheric coherence between motor areas in the alpha band. The effects of rTMS lasted up to 25 min post-stimulation. There was no significant change in EEG-EEG coherence over the hemisphere contralateral to stimulation.

CONCLUSIONS

Low frequency, subthreshold rTMS of the motor cortex increases ipsilateral cortico-cortical and interhemispheric coherence in the alpha band. This may, in part, mediate the inhibitory effects of low frequency rTMS.

Neural mechanisms for generating rate and temporal codes in model CA3 pyramidal cells

The effect of synaptic inhibition on burst firing of a two-compartment model of a CA3 pyramidal cell is considered. We show that, depending on its timing, a short dose of fast decaying synaptic inhibition can either delay or advance the timing of firing of subsequent bursts. Moreover, increasing the strength of the inhibitory input is shown to modulate the burst profile from a full complex burst, to a burst with multiple spikes, to single spikes. We additionally show how slowly decaying inhibitory input can be used to synchronize a network of pyramidal cells. Implications for the phase precession phenomenon of hippocampal place cells and for the generation of temporal and rate codes are discussed.

Localized bumps of activity sustained by inhibition in a two-layer thalamic network

Based on head direction experiments in rats, the existence of localized bumps of thalamic activity has been proposed. We computationally demonstrate the existence of a novel class of localized bump solutions in a two-layer conductance-based thalamic network and analyze the mechanisms behind these stable patterns. In contrast to previous models of bump activity, here inhibition plays a crucial role in initially spreading neuronal firing and in subsequently sustaining it. In our model, we incorporate local strong, fast GABA(A) inhibition and diffuse weak, slow GABA(B) inhibition, based on previous biophysical experiments. These forms of inhibition contribute in different, yet complementary, ways to the observed pattern formation.

Role of synaptic delay in organizing the behavior of networks of self-inhibiting neurons

We consider a pair of mutually coupled inhibitory neurons in which each neuron is also self-inhibitory. We show that the size of the synaptic delay determines the existence and stability of solutions. For small delays, there is no synchronous solution, but a stable antiphase and a stable on-state solution. For long delays, only the synchronous solution is stable. For intermediate delays, either the antiphase or synchronous solutions are stable. In contrast to prior work, for stability of synchrony, we only require the existence of a single slow process.

Alpha-frequency rhythms desynchronize over long cortical distances: a modeling study

Neocortical networks of excitatory and inhibitory neurons can display alpha(a)-frequency rhythms when an animal is in a resting or unfocused state. Unlike some gamma- and beta-frequency rhythms, experimental observations in cats have shown that these alpha-frequency rhythms need not synchronize over long cortical distances. Here, we develop a network model of synaptically coupled excitatory and inhibitory cells to study this asynchrony. The cells of the local circuit are modeled on the neurons found in layer V of the neocortex where alpha-frequency rhythms are thought to originate. Cortical distance is represented by a pair of local circuits coupled with a delay in synaptic propagation. Mathematical analysis of this model reveals that the h and T currents present in layer V pyramidal (excitatory) cells not only produce and regulate the alpha-frequency rhythm but also lead to the occurrence of spatial asynchrony. In particular, these inward currents cause excitation and inhibition to have nonintuitive effects in the network, with excitation delaying and inhibition advancing the firing time of cells; these reversed effects create the asynchrony. Moreover, increased excitatory to excitatory connections can lead to further desynchronization. However, the local rhythms have the property that, in the absence of excitatory to excitatory connections, if the participating cells are brought close to synchrony (for example, by common input), they will remain close to synchrony for a substantial time.

Modeling alternation to synchrony with inhibitory coupling: a neuromorphic VLSI approach

We developed an analog very large-scale integrated system of two mutually inhibitory silicon neurons that display several different stable oscillations. For example, oscillations can be synchronous with weak inhibitory coupling and alternating with relatively strong inhibitory coupling. All oscillations observed experimentally were predicted by bifurcation analysis of a corresponding mathematical model. The synchronous oscillations do not require special synaptic properties and are apparently robust enough to survive the variability and constraints inherent in this physical system. In biological experiments with oscillatory neuronal networks, blockade of inhibitory synaptic coupling can sometimes lead to synchronous oscillations. An example of this phenomenon is the transition from alternating to synchronous bursting in the swimming central pattern generator of lamprey when synaptic inhibition is blocked by strychnine. Our results suggest a simple explanation for the observed oscillatory transitions in the lamprey central pattern generator network: that inhibitory connectivity alone is sufficient to produce the observed transition.

A temporal mechanism for generating the phase precession of hippocampal place cells

The phase relationship between the activity of hippocampal place cells and the hippocampal theta rhythm systematically processes as the animal runs through the region in an environment called the place field of the cell. We present a minimal biophysical model of the phase precession of place cells in region CA3 of the hippocampus. The model describes the dynamics of two coupled point neurons--namely, a pyramidal cell and an interneuron, the latter of which is driven by a pacemaker input. Outside of the place field, the network displays a stable, background firing pattern that is locked to the theta rhythm. The pacemaker input drives the interneuron, which in turn activates the pyramidal cell. A single stimulus to the pyramidal cell from the dentate gyrus, simulating entrance into the place field, reorganizes the functional roles of the cells in the network for a number of cycles of the theta rhythm. In the reorganized network, the pyramidal cell drives the interneuron at a higher frequency than the theta frequency, thus causing a systematic precession relative to the theta input. The frequency of the pyramidal cell can vary to account for changes in the animal's running speed. The transient dynamics end after up to 360 degrees of phase precession when the pacemaker input to the interneuron occurs at a phase to return the network to the stable background firing pattern, thus signaling the end of the place field. Our model, in contrast to others, reports that phase precession is a temporally, and not spatially, controlled process. We also predict that like pyramidal cells, interneurons phase precess. Our model provides a mechanism for shutting off place cell firing after the animal has crossed the place field, and it explains the observed nearly 360 degrees of phase precession. We also describe how this model is consistent with a proposed autoassociative memory role of the CA3 region.