David Golomb

Clustering in globally coupled inhibitory neurons

Abstract A model of a large population of identical excitable neurons with a global slowly decaying inhibitory coupling is studied and its patterns of synchrony are examined. In addition to converging to a homogeneous fixed point and a homogeneous limit cycle, the system exhibits cluster states, in which it breaks spontaneously into a few macroscopically big clusters, each of which is fully synchronized. A method for calculating the stability of cluster states is described and used for investigating the dynamical behavior of the network versus the parameters that describe the neurons and synapses. Effects of stochastic noise on the network dynamics are discussed. At large enough noise the system goes to a globally stationary state. Low levels of noise preserve the cluster-like neuron trajectories. In the regime where a noiseless system converges to a fully synchronized periodic state, relatively low noise levels cause neurons to burst only every two or more consecutive time periods.

The Number of Synaptic Inputs and the Synchrony of Large, Sparse Neuronal Networks

The prevalence of coherent oscillations in various frequency ranges in the central nervous system raises the question of the mechanisms that synchronize large populations of neurons. We study synchronization in models of large networks of spiking neurons with random sparse connectivity. Synchrony occurs only when the average number of synapses, M, that a cell receives is larger than a critical value, Mc. Below Mc, the system is in an asynchronous state. In the limit of weak coupling, assuming identical neurons, we reduce the model to a system of phase oscillators that are coupled via an effective interaction, . In this framework, we develop an approximate theory for sparse networks of identical neurons to estimate Mc analytically from the Fourier coefficients of . Our approach relies on the assumption that the dynamics of a neuron depend mainly on the number of cells that are presynaptic to it. We apply this theory to compute Mc for a model of inhibitory networks of integrate-and-fire (I&F) neurons as a function of the intrinsic neuronal properties (e.g., the refractory period Tr), the synaptic time constants, and the strength of the external stimulus, Iext. The number Mc is found to be nonmonotonous with the strength of Iext. For Tr 0, we estimate the minimum value of Mc over all the parameters of the model to be 363.8. Above Mc, the neurons tend to fire in smeared one-cluster states at high firing rates and smeared two-or-more-cluster states at low firing rates. Refractoriness decreases Mc at intermediate and high firing rates. These results are compared to numerical simulations. We show numerically that systems with different sizes, N, behave in the same way provided the connectivity, M, is such that 1/Meff 1/M 1/N remains constant when N varies. This allows extrapolating the large N behavior of a network from numerical simulations of networks of relatively small sizes (N = 800 in our case). We find that our theory predicts with remarkable accuracy the value of Mc and the patterns of synchrony above Mc, provided the synaptic coupling is not too large. We also study the strong coupling regime of inhibitory sparse networks. All of our simulations demonstrate that increasing the coupling strength reduces the level of synchrony of the neuronal activity. Above a critical coupling strength, the network activity is asynchronous. We point out a fundamental limitation for the mechanisms of synchrony relying on inhibition alone, if heterogeneities in the intrinsic properties of the neurons and spatial fluctuations in the external input are also taken into account.

The Combined Effects of Inhibitory and Electrical Synapses in Synchrony

Recent experimental results have shown that GABAergic interneurons in the central nervous system are frequently connected via electrical synapses. Hence, depending on the area or the subpopulation, interneurons interact via inhibitory synapses or electrical synapses alone or via both types of interactions. The theoretical work presented here addresses the significance of these different modes of interactions for the interneuron networks dynamics. We consider the simplest system in which this issue can be investigated in models or in experiments: a pair of neurons, interacting via electrical synapses, inhibitory synapses, or both, and activated by the injection of a noisy external current. Assuming that the couplings and the noise are weak, we derive an analytical expression relating the cross-correlation (CC) of the activity of the two neurons to the phase response function of the neurons. When electrical and inhibitory interactions are not too strong, they combine their effect in a linear manner. In this regime, the effect of electrical and inhibitory interactions when combined can be deduced knowing the effects of each of the interactions separately. As a consequence, depending on intrinsic neuronal proper-ties, electrical and inhibitory synapses may cooperate, both promoting synchrony, or may compete, with one promoting synchrony while the other impedes it. In contrast, for sufficiently strong couplings, the two types of synapses combine in a nonlinear fashion. Remarkably, we find that in this regime, combining electrical synapses with inhibition ampli-fies synchrony, whereas electrical synapses alone would desynchronize the activity of the neurons. We apply our theory to predict how the shape of the CC of two neurons changes as a function of ionic channel conduc-tances, focusing on the effect of persistent sodium conductance, of the firing rate of the neurons and the nature and the strength of their interac-tions. These predictions may be tested using dynamic clamp techniques.

How well can we estimate the information carried in neuronal responses from limited samples

It is difficult to extract the information carried by neuronal responses about a set of stimuli because limited data samples result in biased es timates. Recently two improved procedures have been developed to calculate information from experimental results: a binning-and-correcting procedure and a neural network procedure. We have used data produced from a model of the spatiotemporal receptive fields of parvocellular and magnocellular lateral geniculate neurons to study the performance of these methods as a function of the number of trials used. Both procedures yield accurate results for one-dimensional neuronal codes. They can also be used to produce a reasonable estimate of the extra information in a three-dimensional code, in this instance, within 0.05-0.1 bit of the asymptotically calculated valueabout 10 of the total transmitted information. We believe that this performance is much more accurate than previous procedures.

Muscle Architecture in the Mystacial Pad of the Rat

The vibrissal system of the rat is an example of active tactile sensing, and has recently been used as a prototype in construction of touch‐oriented robots. Active vibrissal exploration and touch are enabled and controlled by musculature of the mystacial pad. So far, knowledge about motor control of the rat vibrissal system has been extracted from what is known about the vibrissal systems of other species, mainly mice and hamsters, since a detailed description of the musculature of the rat mystacial pad was lacking. In the present work, the musculature of the rat mystacial pad was revealed by slicing the mystacial pad in four different planes, staining of mystacial pad slices for cytochrome oxidase, and tracking spatial organization of mystacial pad muscles in consecutive slices. We found that the rat mystacial pad contains four superficial extrinsic muscles and five parts of the M. nasolabialis profundus. The connection scheme of the three parts of the M. nasolabialis profundus is described here for the first time. These muscles are inserted into the plate of the mystacial pad, and thus, their contraction causes whisker retraction. All the muscles of the rat mystacial pad contained three types of skeletal striated fibers (red, white, and intermediate). Although the entire rat mystacial pad usually functions as unity, our data revealed its structural segmentation into nasal and maxillary subdivisions. The mechanisms of whisking in the rat, and hypotheses concerning biomechanical interactions during whisking, are discussed with respect to the muscle architecture of the rat mystacial pad. Anat Rec 293:1192–1206, 2010.

Tapered whiskers are required for active tactile sensation

Many mammals forage and burrow in dark constrained spaces. Touch through facial whiskers is important during these activities, but the close quarters makes whisker deployment challenging. The diverse shapes of facial whiskers reflect distinct ecological niches. Rodent whiskers are conical, often with a remarkably linear taper. Here we use theoretical and experimental methods to analyze interactions of mouse whiskers with objects. When pushed into objects, conical whiskers suddenly slip at a critical angle. In contrast, cylindrical whiskers do not slip for biologically plausible movements. Conical whiskers sweep across objects and textures in characteristic sequences of brief sticks and slips, which provide information about the tactile world. In contrast, cylindrical whiskers stick and remain stuck, even when sweeping across fine textures. Thus the conical whisker structure is adaptive for sensor mobility in constrained environments and in feature extraction during active haptic exploration of objects and surfaces. DOI: http://dx.doi.org/10.7554/eLife.01350.001

Temporal and spatial characteristics of vibrissa responses to motor commands.

A mechanistic description of the generation of whisker movements is essential for understanding the control of whisking and vibrissal active touch. We explore how facial-motoneuron spikes are translated, via an intrinsic muscle, to whisker movements. This is achieved by constructing, simulating, and analyzing a computational, biomechanical model of the motor plant, and by measuring spiking to movement transformations at small and large angles using high-precision whisker tracking in vivo. Our measurements revealed a supralinear summation of whisker protraction angles in response to consecutive motoneuron spikes with moderate interspike intervals (5 ms < Δt < 30 ms). This behavior is explained by a nonlinear transformation from intracellular changes in Ca2+ concentration to muscle force. Our model predicts the following spatial constraints: (1) Contraction of a single intrinsic muscle results in movement of its two attached whiskers with different amplitudes; the relative amplitudes depend on the resting angles and on the attachment location of the intrinsic muscle on the anterior whisker. Counterintuitively, for a certain range of resting angles, activation of a single intrinsic muscle can lead to a retraction of one of its two attached whiskers. (2) When a whisker is pulled by its two adjacent muscles with similar forces, the protraction amplitude depends only weakly on the resting angle. (3) Contractions of two adjacent muscles sums up linearly for small amplitudes and supralinearly for larger amplitudes. The model provides a direct translation from motoneuron spikes to whisker movements and can serve as a building block in closed-loop motor–sensory models of active touch.

Effects of delay on the type and velocity of travelling pulses in neuronal networks with spatially decaying connectivity.

We study a one-dimensional model of integrate-and-fire neurons that are allowed to fire only one spike, and are coupled by excitatory synapses with delay. At small delay values, this model describes a disinhibited cortical slice. At large delay values, the model is a reduction of a model of thalamic networks composed of excitatory and inhibitory neurons, in which the excitatory neurons show the post-inhibitory rebound mechanism. The velocity and stability of propagating continuous pulses are calculated analytically. Two pulses with different velocities exist if the synaptic coupling is larger than a minimal value; the pulse with the lower velocity is always unstable. Above a certain critical value of the constant delay, continuous pulses lose stability via a Hopf bifurcation, and lurching pulses emerge. The parameter regime for which lurching occurs is strongly affected by the synaptic footprint (connectivity) shape. A bistable regime, in which both continuous and lurching pulses can propagate, may occur with square or Gaussian footprint shapes but not with an exponential footprint shape. A perturbation calculation is used in order to calculate the spatial lurching period and the velocity of lurching pulses at large delay values. For strong synaptic coupling, the velocity of the lurching pulse is governed by the tail of the synaptic footprint shape. Moreover, the velocities of continuous and lurching pulses have the same functional dependencies on the strength of the synaptic coupling strength gsyn: they increase logarithmically with gsyn for an exponential footprint shape, they scale like (ln gsyn)1/2 for a Gaussian footprint shape, and they are bounded for a square footprint shape or any shape with a finite support. We find analytically how the axonal propagation velocity reduces the velocity of continuous pulses; it does not affect the critical delay. We conclude that the differences in velocity and shape between the front of thalamic spindle waves in vitro and cortical paroxysmal discharges stem from their different effective delays.

Dorsorostral snout muscles in the rat subserve coordinated movement for whisking and sniffing.

Histochemical examination of the dorsorostral quadrant of the rat snout revealed superficial and deep muscles that are involved in whisking, sniffing, and airflow control. The part of M. nasolabialis profundus that acts as an intrinsic (follicular) muscle to facilitate protraction and translation of the vibrissae is described. An intraturbinate and selected rostral‐most nasal muscles that can influence major routs of inspiratory airflow and rhinarial touch through their control of nostril configuration, atrioturbinate and rhinarium position, were revealed. Anat Rec, 2012.

Period Doubling of Calcium Spike Firing in a Model of a Purkinje Cell Dendrite

Recordings from cerebellar Purkinje cell dendrites have revealed that in response to sustained current injection, the cell firing pattern can move from tonic firing of Ca2+ spikes to doublet firing and even to quadruplet firing or more complex firing. These firing patterns are not modified substantially if Na+ currents are blocked. We show that the experimental results can be viewed as a slow transition of the neuronal dynamics through a period-doubling bifurcation. To further support this conclusion and to understand the underlying mechanism that leads to doublet firing, we develop and study a simple, one-compartment model of Purkinje cell dendrite. The neuron can also exhibit quadruplet and chaotic firing patterns that are similar to the firing patterns that some of the Purkinje cells exhibit experimentally. The effects of parameters such as temperature, applied current, and potassium reversal potential in the model resemble their effects in experiments. The model dynamics involve three time scales. Ca2+- dependent K+ currents, with intermediate time scales, are responsible for the appearance of doublet firing, whereas a very slow hyperpolarizing current transfers the neuron from tonic to doublet firing. We use the fast-slow analysis to separate the effects of the three time scales. Fast-slow analysis of the neuronal dynamics, with the activation variable of the very slow, hyperpolarizing current considered as a parameter, reveals that the transitions occurs via a cascade of period-doubling bifurcations of the fast and intermediate subsystem as this slow variable increases. We carry out another analysis, with the Ca2+ concentration considered as a parameter, to investigate the conditions for the generation of doublet firing in systems with one effective variable with intermediate time scale, in which the rest state of the fast subsystem is terminated by a saddle-node bifurcation. We find that the scenario of period doubling in these systems can occur only if (1) the time scale of the intermediate variable (here, the decay rate of the calcium concentration) is slow enough in comparison with the interspike interval of the tonic firing at the transition but is not too slow and (2) there is a bistability of the fast subsystem of the spike-generating variables.