Average activity of excitatory and inhibitory neural populations

Pulse Shape and Voltage-Dependent Synchronization in Spiking Neuron Networks

Pulse-coupled spiking neural networks are a powerful tool to gain mechanistic insights into how neurons self-organize to produce coherent collective behavior. These networks use simple spiking neuron models, such as the θ-neuron or the quadratic integrate-and-fire (QIF) neuron, that replicate the essential features of real neural dynamics. Interactions between neurons are modeled with infinitely narrow pulses, or spikes, rather than the more complex dynamics of real synapses. To make these networks biologically more plausible, it has been proposed that they must also account for the finite width of the pulses, which can have a significant impact on the network dynamics. However, the derivation and interpretation of these pulses are contradictory, and the impact of the pulse shape on the network dynamics is largely unexplored. Here, I take a comprehensive approach to pulse coupling in networks of QIF and θ-neurons. I argue that narrow pulses activate voltage-dependent synaptic conductances and show how to implement them in QIF neurons such that their effect can last through the phase after the spike. Using an exact low-dimensional description for networks of globally coupled spiking neurons, I prove for instantaneous interactions that collective oscillations emerge due to an effective coupling through the mean voltage. I analyze the impact of the pulse shape by means of a family of smooth pulse functions with arbitrary finite width and symmetric or asymmetric shapes. For symmetric pulses, the resulting voltage coupling is not very effective in synchronizing neurons, but pulses that are slightly skewed to the phase after the spike readily generate collective oscillations. The results unveil a voltage-dependent spike synchronization mechanism at the heart of emergent collective behavior, which is facilitated by pulses of finite width and complementary to traditional synaptic transmission in spiking neuron networks.

Exact Mean-Field Theory Explains the Dual Role of Electrical Synapses in Collective Synchronization

Electrical synapses play a major role in setting up neuronal synchronization, but the precise mechanisms whereby these synapses contribute to synchrony are subtle and remain elusive. To investigate these mechanisms mean-field theories for quadratic integrate-and-fire neurons with electrical synapses have been recently put forward. Still, the validity of these theories is controversial since they assume that the neurons produce unrealistic, symmetric spikes, ignoring the well-known impact of spike shape on synchronization. Here, we show that the assumption of symmetric spikes can be relaxed in such theories. The resulting mean-field equations reveal a dual role of electrical synapses: First, they equalize membrane potentials favoring the emergence of synchrony. Second, electrical synapses act as "virtual chemical synapses," which can be either excitatory or inhibitory depending upon the spike shape. Our results offer a precise mathematical explanation of the intricate effect of electrical synapses in collective synchronization. This reconciles previous theoretical and numerical works, and confirms the suitability of recent low-dimensional mean-field theories to investigate electrically coupled neuronal networks.

The Winfree model with non-infinitesimal phase-response curve: Ott-Antonsen theory

A novel generalization of the Winfree model of globally coupled phase oscillators, representing phase reduction under finite coupling, is studied analytically. We consider interactions through a non-infinitesimal (or finite) phase-response curve (PRC), in contrast to the infinitesimal PRC of the original model. For a family of non-infinitesimal PRCs, the global dynamics is captured by one complex-valued ordinary differential equation resorting to the Ott-Antonsen ansatz. The phase diagrams are thereupon obtained for four illustrative cases of non-infinitesimal PRC. Bistability between collective synchronization and full desynchronization is observed in all cases.

Dynamical model for the neural activity of singing Serinus canaria

Vocal production in songbirds is a key topic regarding the motor control of a complex, learned behavior. Birdsong is the result of the interaction between the activity of an intricate set of neural nuclei specifically dedicated to song production and learning (known as the "song system"), the respiratory system and the vocal organ. These systems interact and give rise to precise biomechanical motor gestures which result in song production. Telencephalic neural nuclei play a key role in the production of motor commands that drive the periphery, and while several attempts have been made to understand their coding strategy, difficulties arise when trying to understand neural activity in the frame of the song system as a whole. In this work, we report neural additive models embedded in an architecture compatible with the song system to provide a tool to reduce the dimensionality of the problem by considering the global activity of the units in each neural nucleus. This model is capable of generating outputs compatible with measurements of air sac pressure during song production in canaries (Serinus canaria). In this work, we show that the activity in a telencephalic nucleus required by the model to reproduce the observed respiratory gestures is compatible with electrophysiological recordings of single neuron activity in freely behaving animals.

Exact firing rate model reveals the differential effects of chemical versus electrical synapses in spiking networks

Chemical and electrical synapses shape the dynamics of neuronal networks. Numerous theoretical studies have investigated how each of these types of synapses contributes to the generation of neuronal oscillations, but their combined effect is less understood. This limitation is further magnified by the impossibility of traditional neuronal mean-field models-also known as firing rate models or firing rate equations-to account for electrical synapses. Here, we introduce a firing rate model that exactly describes the mean-field dynamics of heterogeneous populations of quadratic integrate-and-fire (QIF) neurons with both chemical and electrical synapses. The mathematical analysis of the firing rate model reveals a well-established bifurcation scenario for networks with chemical synapses, characterized by a codimension-2 cusp point and persistent states for strong recurrent excitatory coupling. The inclusion of electrical coupling generally implies neuronal synchrony by virtue of a supercritical Hopf bifurcation. This transforms the cusp scenario into a bifurcation scenario characterized by three codimension-2 points (cusp, Takens-Bogdanov, and saddle-node separatrix loop), which greatly reduces the possibility for persistent states. This is generic for heterogeneous QIF networks with both chemical and electrical couplings. Our results agree with several numerical studies on the dynamics of large networks of heterogeneous spiking neurons with electrical and chemical couplings.

Complexity reduction ansatz for systems of interacting orientable agents: Beyond the Kuramoto model

Previous results have shown that a large class of complex systems consisting of many interacting heterogeneous phase oscillators exhibit an attracting invariant manifold. This result has enabled reduced analytic system descriptions from which all the long term dynamics of these systems can be calculated. Although very useful, these previous results are limited by the restriction that the individual interacting system components have one-dimensional dynamics, with states described by a single, scalar, angle-like variable (e.g., the Kuramoto model). In this paper, we consider a generalization to an appropriate class of coupled agents with higher-dimensional dynamics. For this generalized class of model systems, we demonstrate that the dynamics again contain an invariant manifold, hence enabling previously inaccessible analysis and improved numerical study, allowing a similar simplified description of these systems. We also discuss examples illustrating the potential utility of our results for a wide range of interesting situations.

Observable for a Large System of Globally Coupled Excitable Units

The study of large arrays of coupled excitable systems has largely benefited from a technique proposed by Ott and Antonsen, which results in a low dimensional system of equations for the system’s order parameter. In this work, we show how to explicitly introduce a variable describing the global synaptic activation of the network into these family of models. This global variable is built by adding realistic synaptic time traces. We propose that this variable can, under certain conditions, be a good proxy for the local field potential of the network. We report experimental, in vivo, electrophysiology data supporting this claim.

A multiple timescales approach to bridging spiking- and population-level dynamics

A rigorous bridge between spiking-level and macroscopic quantities is an on-going and well-developed story for asynchronously firing neurons, but focus has shifted to include neural populations exhibiting varying synchronous dynamics. Recent literature has used the Ott-Antonsen ansatz (2008) to great effect, allowing a rigorous derivation of an order parameter for large oscillator populations. The ansatz has been successfully applied using several models including networks of Kuramoto oscillators, theta models, and integrate-and-fire neurons, along with many types of network topologies. In the present study, we take a converse approach: given the mean field dynamics of slow synapses, we predict the synchronization properties of finite neural populations. The slow synapse assumption is amenable to averaging theory and the method of multiple timescales. Our proposed theory applies to two heterogeneous populations of N excitatory n-dimensional and N inhibitory m-dimensional oscillators with homogeneous synaptic weights. We then demonstrate our theory using two examples. In the first example, we take a network of excitatory and inhibitory theta neurons and consider the case with and without heterogeneous inputs. In the second example, we use Traub models with calcium for the excitatory neurons and Wang-Buzsáki models for the inhibitory neurons. We accurately predict phase drift and phase locking in each example even when the slow synapses exhibit non-trivial mean-field dynamics.

Firing rate equations require a spike synchrony mechanism to correctly describe fast oscillations in inhibitory networks

Recurrently coupled networks of inhibitory neurons robustly generate oscillations in the gamma band. Nonetheless, the corresponding Wilson-Cowan type firing rate equation for such an inhibitory population does not generate such oscillations without an explicit time delay. We show that this discrepancy is due to a voltage-dependent spike-synchronization mechanism inherent in networks of spiking neurons which is not captured by standard firing rate equations. Here we investigate an exact low-dimensional description for a network of heterogeneous canonical Class 1 inhibitory neurons which includes the sub-threshold dynamics crucial for generating synchronous states. In the limit of slow synaptic kinetics the spike-synchrony mechanism is suppressed and the standard Wilson-Cowan equations are formally recovered as long as external inputs are also slow. However, even in this limit synchronous spiking can be elicited by inputs which fluctuate on a time-scale of the membrane time-constant of the neurons. Our meanfield equations therefore represent an extension of the standard Wilson-Cowan equations in which spike synchrony is also correctly described.

Population models describing the average activity of large neuronal ensembles are a powerful mathematical tool to investigate the principles underlying cooperative function of large neuronal systems. However, these models do not properly describe the phenomenon of spike synchrony in networks of neurons. In particular, they fail to capture the onset of synchronous oscillations in networks of inhibitory neurons. We show that this limitation is due to a voltage-dependent synchronization mechanism which is naturally present in spiking neuron models but not captured by traditional firing rate equations. Here we investigate a novel set of macroscopic equations which incorporate both firing rate and membrane potential dynamics, and that correctly generate fast inhibition-based synchronous oscillations. In the limit of slow-synaptic processing oscillations are suppressed, and the model reduces to an equation formally equivalent to the Wilson-Cowan model.

Connecting the Kuramoto Model and the Chimera State

Since its discovery in 2002, the chimera state has frequently been described as a counterintuitive, puzzling phenomenon. The Kuramoto model, in contrast, has become a celebrated paradigm useful for understanding a range of phenomena related to phase transitions, synchronization, and network effects. Here we show that the chimera state can be understood as emerging naturally through a symmetry-breaking bifurcation from the Kuramoto model's partially synchronized state. Our analysis sheds light on recent observations of chimera states in laser arrays, chemical oscillators, and mechanical pendula.

Nonlinear dynamics in the study of birdsong

Birdsong, a rich and complex behavior, is a stellar model to understand a variety of biological problems, from motor control to learning. It also enables us to study how behavior emerges when a nervous system, a biomechanical device and the environment interact. In this review, I will show that many questions in the field can benefit from the approach of nonlinear dynamics, and how birdsong can inspire new directions for research in dynamics.

From perception to action in songbird production: dynamics of a whole loop

Birdsong emerges when a set of highly interconnected brain areas manage to generate a complex output. This consists of precise respiratory rhythms as well as motor instructions to control the vocal organ configuration. In this way, during birdsong production, dedicated cortical areas interact with life-supporting ones in the brainstem, such as the respiratory nuclei. We discuss an integrative view of this interaction together with a widely accepted "top-down" representation of the song system. We also show that a description of this neural network in terms of dynamical systems allows to explore songbird production and processing by generating testable predictions.

Complex behavior in chains of nonlinear oscillators

This article outlines sufficient conditions under which a one-dimensional chain of identical nonlinear oscillators can display complex spatio-temporal behavior. The units are described by phase equations and consist of excitable oscillators. The interactions are local and the network is poised to a critical state by balancing excitation and inhibition locally. The results presented here suggest that in networks composed of many oscillatory units with local interactions, excitability together with balanced interactions is sufficient to give rise to complex emergent features. For values of the parameters where complex behavior occurs, the system also displays a high-dimensional bifurcation where an exponentially large number of equilibria are borne in pairs out of multiple saddle-node bifurcations.

Modeling the network dynamics of pulse-coupled neurons

We derive a mean-field approximation for the macroscopic dynamics of large networks of pulse-coupled theta neurons in order to study the effects of different network degree distributions and degree correlations (assortativity). Using the ansatz of Ott and Antonsen [Chaos 18, 037113 (2008)], we obtain a reduced system of ordinary differential equations describing the mean-field dynamics, with significantly lower dimensionality compared with the complete set of dynamical equations for the system. We find that, for sufficiently large networks and degrees, the dynamical behavior of the reduced system agrees well with that of the full network. This dimensional reduction allows for an efficient characterization of system phase transitions and attractors. For networks with tightly peaked degree distributions, the macroscopic behavior closely resembles that of fully connected networks previously studied by others. In contrast, networks with highly skewed degree distributions exhibit different macroscopic dynamics due to the emergence of degree dependent behavior of different oscillators. For nonassortative networks (i.e., networks without degree correlations), we observe the presence of a synchronously firing phase that can be suppressed by the presence of either assortativity or disassortativity in the network. We show that the results derived here can be used to analyze the effects of network topology on macroscopic behavior in neuronal networks in a computationally efficient fashion.

The Effect of Gap Junctional Coupling on the Spatiotemporal Patterns of Ca2+ Signals and the Harmonization of Ca2+-Related Cellular Responses

Calcium ions (Ca²⁺) are important mediators of a great variety of cellular activities e.g. in response to an agonist activation of a receptor. The magnitude of a cellular response is often encoded by frequency modulation of Ca²⁺ oscillations and correlated with the stimulation intensity. The stimulation intensity highly depends on the sensitivity of a cell to a certain agonist. In some cases, it is essential that neighboring cells produce a similar and synchronized response to an agonist despite their different sensitivity. In order to decipher the presumed function of Ca²⁺ waves spreading among connecting cells, a mathematical model was developed. This model allows to numerically modifying the connectivity probability between neighboring cells, the permeability of gap junctions and the individual sensitivity of cells to an agonist. Here, we show numerically that strong gap junctional coupling between neighbors ensures an equilibrated response to agonist stimulation via formation of Ca²⁺ phase waves, i.e. a less sensitive neighbor will produce the same or similar Ca²⁺ signal as its highly sensitive neighbor. The most sensitive cells within an ensemble are the wave initiator cells. The Ca²⁺ wave in the cytoplasm is driven by a sensitization wave front in the endoplasmic reticulum. The wave velocity is proportional to the cellular sensitivity and to the strength of the coupling. The waves can form different patterns including circular rings and spirals. The observed pattern depends on the strength of noise, gap junctional permeability and the connectivity probability between neighboring cells. Our simulations reveal that one highly sensitive region gradually takes the lead within the entire noisy system by generating directed circular phase waves originating from this region.

The calcium ion (Ca²⁺), a universal signaling molecule, is widely recognized to play a fundamental role in the regulation of various biological processes. Agonist–evoked Ca²⁺ signals often manifest as rhythmic changes in the cytosolic free Ca²⁺ concentration (c_cyt) called Ca²⁺ oscillations. Stimuli intensity was found to be proportional to the oscillation frequency and the evoked down-steam cellular response. Stochastic receptor expression in individual cells in a cell population inevitably leads to individually different oscillation frequencies and individually different Ca²⁺-related cellular responses. However, in many organs, the neighboring cells have to overcome their individually different sensitivity and produce a synchronized response. Gap junctions are integral membrane structures that enable the direct cytoplasmic exchange of Ca²⁺ ions and InsP₃ molecules between neighboring cells. By simulations, we were able to demonstrate how the strength of intercellular gap junctional coupling in relation to stimulus intensity can modify the spatiotemporal patterns of Ca²⁺ signals and harmonize the Ca²⁺-related cellular responses via synchronization of oscillation frequency. We demonstrate that the most sensitive cells are the wave initiator cells and that a highly sensitive region plays an important role in the determination of the Ca²⁺ phase wave direction. This sensitive region will then also progressively determine the global behavior of the entire system.