Frequency control in synchronized networks of inhibitory neurons

We analyze the control of frequency for a synchronized inhibitory neuronal network. The analysis is done for a reduced membrane model with a biophysically-based synaptic influence. We argue that such a reduced model can quantitatively capture the frequency behavior of a larger class of neuronal models. We show that in different parameter regimes, the network frequency depends in different ways on the intrinsic and synaptic time constants. Only in one portion of the parameter space, called `phasic', is the network period proportional to the synaptic decay time. These results are discussed in connection with previous work of the authors, which showed that for mildly heterogeneous networks, the synchrony breaks down, but coherence is preserved much more for systems in the phasic regime than in the other regimes. These results imply that for mildly heterogeneous networks, the existence of a coherent rhythm implies a linear dependence of the network period on synaptic decay time, and a much weaker dependence on the drive to the cells. We give experimental evidence for this conclusion.Comment: 18 pages, 3 figures, Kluwer.sty. J. Comp. Neurosci. (in press). Originally submitted to the neuro-sys archive which was never publicly announced (was 9803001

Synchronization and oscillatory dynamics in heterogeneous mutually inhibited neurons

We study some mechanisms responsible for synchronous oscillations and loss of synchrony at physiologically relevant frequencies (10-200 Hz) in a network of heterogeneous inhibitory neurons. We focus on the factors that determine the level of synchrony and frequency of the network response, as well as the effects of mild heterogeneity on network dynamics. With mild heterogeneity, synchrony is never perfect and is relatively fragile. In addition, the effects of inhibition are more complex in mildly heterogeneous networks than in homogeneous ones. In the former, synchrony is broken in two distinct ways, depending on the ratio of the synaptic decay time to the period of repetitive action potentials ($\tau_s/T$), where $T$ can be determined either from the network or from a single, self-inhibiting neuron. With $\tau_s/T > 2$, corresponding to large applied current, small synaptic strength or large synaptic decay time, the effects of inhibition are largely tonic and heterogeneous neurons spike relatively independently. With $\tau_s/T < 1$, synchrony breaks when faster cells begin to suppress their less excitable neighbors; cells that fire remain nearly synchronous. We show numerically that the behavior of mildly heterogeneous networks can be related to the behavior of single, self-inhibiting cells, which can be studied analytically.Comment: 17 pages, 6 figures, Kluwer.sty. Journal of Compuational Neuroscience (in press). Originally submitted to the neuro-sys archive which was never publicly announced (was 9802001

Noise-induced synchronization and anti-resonance in excitable systems; Implications for information processing in Parkinson's Disease and Deep Brain Stimulation

We study the statistical physics of a surprising phenomenon arising in large networks of excitable elements in response to noise: while at low noise, solutions remain in the vicinity of the resting state and large-noise solutions show asynchronous activity, the network displays orderly, perfectly synchronized periodic responses at intermediate level of noise. We show that this phenomenon is fundamentally stochastic and collective in nature. Indeed, for noise and coupling within specific ranges, an asymmetry in the transition rates between a resting and an excited regime progressively builds up, leading to an increase in the fraction of excited neurons eventually triggering a chain reaction associated with a macroscopic synchronized excursion and a collective return to rest where this process starts afresh, thus yielding the observed periodic synchronized oscillations. We further uncover a novel anti-resonance phenomenon: noise-induced synchronized oscillations disappear when the system is driven by periodic stimulation with frequency within a specific range. In that anti-resonance regime, the system is optimal for measures of information capacity. This observation provides a new hypothesis accounting for the efficiency of Deep Brain Stimulation therapies in Parkinson's disease, a neurodegenerative disease characterized by an increased synchronization of brain motor circuits. We further discuss the universality of these phenomena in the class of stochastic networks of excitable elements with confining coupling, and illustrate this universality by analyzing various classical models of neuronal networks. Altogether, these results uncover some universal mechanisms supporting a regularizing impact of noise in excitable systems, reveal a novel anti-resonance phenomenon in these systems, and propose a new hypothesis for the efficiency of high-frequency stimulation in Parkinson's disease

The role of inhibitory feedback for information processing in thalamocortical circuits

The information transfer in the thalamus is blocked dynamically during sleep, in conjunction with the occurence of spindle waves. As the theoretical understanding of the mechanism remains incomplete, we analyze two modeling approaches for a recent experiment by Le Masson {\sl et al}. on the thalamocortical loop. In a first step, we use a conductance-based neuron model to reproduce the experiment computationally. In a second step, we model the same system by using an extended Hindmarsh-Rose model, and compare the results with the conductance-based model. In the framework of both models, we investigate the influence of inhibitory feedback on the information transfer in a typical thalamocortical oscillator. We find that our extended Hindmarsh-Rose neuron model, which is computationally less costly and thus siutable for large-scale simulations, reproduces the experiment better than the conductance-based model. Further, in agreement with the experiment of Le Masson {\sl et al}., inhibitory feedback leads to stable self-sustained oscillations which mask the incoming input, and thereby reduce the information transfer significantly.Comment: 16 pages, 15eps figures included. To appear in Physical Review

Rhythms of the nervous system: mathematical themes and variations

The nervous system displays a variety of rhythms in both waking and sleep. These rhythms have been closely associated with different behavioral and cognitive states, but it is still unknown how the nervous system makes use of these rhythms to perform functionally important tasks. To address those questions, it is first useful to understood in a mechanistic way the origin of the rhythms, their interactions, the signals which create the transitions among rhythms, and the ways in which rhythms filter the signals to a network of neurons. This talk discusses how dynamical systems have been used to investigate the origin, properties and interactions of rhythms in the nervous system. It focuses on how the underlying physiology of the cells and synapses of the networks shape the dynamics of the network in different contexts, allowing the variety of dynamical behaviors to be displayed by the same network. The work is presented using a series of related case studies on different rhythms. These case studies are chosen to highlight mathematical issues, and suggest further mathematical work to be done. The topics include: different roles of excitation and inhibition in creating synchronous assemblies of cells, different kinds of building blocks for neural oscillations, and transitions among rhythms. The mathematical issues include reduction of large networks to low dimensional maps, role of noise, global bifurcations, use of probabilistic formulations.Published versio

Tremorgenesis: a new conceptual scheme using reciprocally innervated circuit of neurons

Neural circuits controlling fast movements are inherently unsteady as a result of their reciprocal innervation. This instability is enhanced by increased membrane excitability. Recent studies indicate that the loss of external inhibition is an important factor in the pathogenesis of several tremor disorders such as essential tremor, cerebellar kinetic tremor or parkinsonian tremor. Shaikh and colleagues propose a new conceptual scheme to analyze tremor disorders. Oscillations are simulated by changing the intrinsic membrane properties of burst neurons. The authors use a model neuron of Hodgkin-Huxley type with added hyperpolarization activated cation current (Ih), low threshold calcium current (It), and GABA/glycine mediated chloride currents. Post-inhibitory rebound is taken into account. The model includes a reciprocally innervated circuit of neurons projecting to pairs of agonist and antagonist muscles. A set of four burst neurons has been simulated: inhibitory agonist, inhibitory antagonist, excitatory agonist, and excitatory antagonist. The model fits well with the known anatomical organization of neural circuits for limb movements in premotor/motor areas, and, interestingly, this model does not require any structural modification in the anatomical organization or connectivity of the constituent neurons. The authors simulate essential tremor when Ih is increased. Membrane excitability is augmented by up-regulating Ih and It. A high level of congruence with the recordings made in patients exhibiting essential tremor is reached. These simulations support the hypothesis that increased membrane excitability in potentially unsteady circuits generate oscillations mimicking tremor disorders encountered in daily practice. This new approach opens new perspectives for both the understanding and the treatment of neurological tremor. It provides the rationale for decreasing membrane excitability by acting on a normal ion channel in a context of impaired external inhibition