Dynamics of driven recurrent networks of ON and OFF cells

Novel Functions of Feedback in Electrosensory Processing

Environmental signals act as input and are processed across successive stages in the brain to generate a meaningful behavioral output. However, a ubiquitous observation is that descending feedback projections from more central to more peripheral brain areas vastly outnumber ascending feedforward projections. Such projections generally act to modify how sensory neurons respond to afferent signals. Recent studies in the electrosensory system of weakly electric fish have revealed novel functions for feedback pathways in that their transformation of the afferent input generates neural firing rate responses to sensory signals mediating perception and behavior. In this review, we focus on summarizing these novel and recently uncovered functions and put them into context by describing the more "classical" functions of feedback in the electrosensory system. We further highlight the parallels between the electrosensory system and other systems as well as outline interesting future directions.

Dynamic Control of Synchronous Activity in Networks of Spiking Neurons

Oscillatory brain activity is believed to play a central role in neural coding. Accumulating evidence shows that features of these oscillations are highly dynamic: power, frequency and phase fluctuate alongside changes in behavior and task demands. The role and mechanism supporting this variability is however poorly understood. We here analyze a network of recurrently connected spiking neurons with time delay displaying stable synchronous dynamics. Using mean-field and stability analyses, we investigate the influence of dynamic inputs on the frequency of firing rate oscillations. We show that afferent noise, mimicking inputs to the neurons, causes smoothing of the system’s response function, displacing equilibria and altering the stability of oscillatory states. Our analysis further shows that these noise-induced changes cause a shift of the peak frequency of synchronous oscillations that scales with input intensity, leading the network towards critical states. We lastly discuss the extension of these principles to periodic stimulation, in which externally applied driving signals can trigger analogous phenomena. Our results reveal one possible mechanism involved in shaping oscillatory activity in the brain and associated control principles.

Reduced dynamics for delayed systems with harmonic or stochastic forcing

The analysis of nonlinear delay-differential equations (DDEs) subjected to external forcing is difficult due to the infinite dimensionality of the space in which they evolve. To simplify the analysis of such systems, the present work develops a non-homogeneous center manifold (CM) reduction scheme, which allows the derivation of a time-dependent order parameter equation in finite dimension. This differential equation captures the major dynamical features of the delayed system. The forcing is assumed to be small compared to the amplitude of the autonomous system, in order to cause only small variations of the fixed points and of the autonomous CM. The time-dependent CM is shown to satisfy a non-homogeneous partial differential equation. We first briefly review CM theory for DDEs. Then we show, for the general scalar case, how an ansatz that separates the CM into one for the autonomous problem plus an additional time-dependent order-two correction leads to satisfying results. The paper then details the application to a transcritical bifurcation subjected to single or multiple periodic forcings. The validity limits of the reduction scheme are also highlighted. Finally, we characterize the specific case of additive stochastic driving of the transcritical bifurcation, where additive white noise shifts the mode of the probability density function of the state variable to larger amplitudes.

Bifurcation control of the Morris-Lecar neuron model via a dynamic state-feedback control

In this paper, we address the control problem of bifurcations in the Morris-Lecar (ML) neuron model. With the use of a dynamic state-feedback control, two Hopf bifurcation points in the ML neuron model with Type II excitability can be relocated to new desired locations simultaneously. Also, with the proposed control law, the neuronal excitability characteristics can be transformed from Type I excitability to Type II excitability by changing the type of bifurcation, in which the neuron goes from quiescence to periodic spiking from a saddle node on an invariant circle bifurcation to a Hopf bifurcation. Simulation results are provided.

Neural population densities shape network correlations

The way sensory microcircuits manage cellular response correlations is a crucial question in understanding how such systems integrate external stimuli and encode information. Most sensory systems exhibit heterogeneities in terms of population sizes and features, which all impact their dynamics. This work addresses how correlations between the dynamics of neural ensembles depend on the relative size or density of excitatory and inhibitory populations. To do so, we study an apparently symmetric system of coupled stochastic differential equations that model the evolution of the populations' activities. Excitatory and inhibitory populations are connected by reciprocal recurrent connections, and both receive different stimuli exhibiting a certain level of correlation with each other. A stability analysis is performed, which reveals an intrinsic asymmetry in the distribution of the fixed points with respect to the threshold of the nonlinearities. Based on this, we show how the cross correlation between the population responses depends on the density of the inhibitory population, and that a specific ratio between both population sizes leads to a state of zero correlation. We show that this so-called asynchronous state subsists, despite the presence of stimulus correlation, and most importantly, that it occurs only in asymmetrical systems where one population outnumbers the other. Using linear approximations, we derive analytical expressions for the root of the cross-correlation function and study how the asynchronous state is impacted by the model's parameters. This work suggests a possible explanation for why inhibitory cells outnumber excitatory cells in the visual system.

Neural adaptation facilitates oscillatory responses to static inputs in a recurrent network of ON and OFF cells

We investigate the role of adaptation in a neural field model, composed of ON and OFF cells, with delayed all-to-all recurrent connections. As external spatially profiled inputs drive the network, ON cells receive inputs directly, while OFF cells receive an inverted image of the original signals. Via global and delayed inhibitory connections, these signals can cause the system to enter states of sustained oscillatory activity. We perform a bifurcation analysis of our model to elucidate how neural adaptation influences the ability of the network to exhibit oscillatory activity. We show that slow adaptation encourages input-induced rhythmic states by decreasing the Andronov-Hopf bifurcation threshold. We further determine how the feedback and adaptation together shape the resonant properties of the ON and OFF cell network and how this affects the response to time-periodic input. By introducing an additional frequency in the system, adaptation alters the resonance frequency by shifting the peaks where the response is maximal. We support these results with numerical experiments of the neural field model. Although developed in the context of the circuitry of the electric sense, these results are applicable to any network of spontaneously firing cells with global inhibitory feedback to themselves, in which a fraction of these cells receive external input directly, while the remaining ones receive an inverted version of this input via feedforward di-synaptic inhibition. Thus the results are relevant beyond the many sensory systems where ON and OFF cells are usually identified, and provide the backbone for understanding dynamical network effects of lateral connections and various forms of ON/OFF responses.

Responses of recurrent nets of asymmetric ON and OFF cells

A neural field model of ON and OFF cells with all-to-all inhibitory feedback is investigated. External spatiotemporal stimuli drive the ON and OFF cells with, respectively, direct and inverted polarity. The dynamic differences between networks built of ON and OFF cells ("ON/OFF") and those having only ON cells ("ON/ON") are described for the general case where ON and OFF cells can have different spontaneous firing rates; this asymmetric case is generic. Neural responses to nonhomogeneous static and time-periodic inputs are analyzed in regimes close to and away from self-oscillation. Static stimuli can cause oscillatory behavior for certain asymmetry levels. Time-periodic stimuli expose dynamical differences between ON/OFF and ON/ON nets. Outside the stimulated region, we show that ON/OFF nets exhibit frequency doubling, while ON/ON nets cannot. On the other hand, ON/ON networks show antiphase responses between stimulated and unstimulated regions, an effect that does not rely on specific receptive field circuitry. An analysis of the resonance properties of both net types reveals that ON/OFF nets exhibit larger response amplitude. Numerical simulations of the neural field models agree with theoretical predictions for localized static and time-periodic forcing. This is also the case for simulations of a network of noisy integrate-and-fire neurons. We finally discuss the application of the model to the electrosensory system and to frequency-doubling effects in retina.

Oscillatory response in a sensory network of ON and OFF cells with instantaneous and delayed recurrent connections

A neural field model with multiple cell-to-cell feedback connections is investigated. Our model incorporates populations of ON and OFF cells, receiving sensory inputs with direct and inverted polarity, respectively. Oscillatory responses to spatially localized stimuli are found to occur via Andronov-Hopf bifurcations of stationary activity. We explore the impact of multiple delayed feedback components as well as additional excitatory and/or inhibitory non-delayed recurrent signals on the instability threshold. Paradoxically, instantaneous excitatory recurrent terms are found to enhance network responsiveness by reducing the oscillatory response threshold, allowing smaller inputs to trigger oscillatory activity. Instantaneous inhibitory components do the opposite. The frequency of these response oscillations is further shaped by the polarity of the non-delayed terms.