Synchronization of excitatory neurons with strongly heterogeneous phase responses

Dynamics of Structured Networks of Winfree Oscillators

Winfree oscillators are phase oscillator models of neurons, characterized by their phase response curve and pulsatile interaction function. We use the Ott/Antonsen ansatz to study large heterogeneous networks of Winfree oscillators, deriving low-dimensional differential equations which describe the evolution of the expected state of networks of oscillators. We consider the effects of correlations between an oscillator's in-degree and out-degree, and between the in- and out-degrees of an “upstream” and a “downstream” oscillator (degree assortativity). We also consider correlated heterogeneity, where some property of an oscillator is correlated with a structural property such as degree. We finally consider networks with parameter assortativity, coupling oscillators according to their intrinsic frequencies. The results show how different types of network structure influence its overall dynamics.

Effect of Phase Response Curve Shape and Synaptic Driving Force on Synchronization of Coupled Neuronal Oscillators

The role of the phase response curve (PRC) shape on the synchrony of synaptically coupled oscillating neurons is examined. If the PRC is independent of the phase, because of the synaptic form of the coupling, synchrony is found to be stable for both excitatory and inhibitory coupling at all rates, whereas the antisynchrony becomes stable at low rates. A faster synaptic rise helps extend the stability of antisynchrony to higher rates. If the PRC is not constant but has a profile like that of a leaky integrate-and-fire model, then, in contrast to the earlier reports that did not include the voltage effects, mutual excitation could lead to stable synchrony provided the synaptic reversal potential is below the voltage level the neuron would have reached in the absence of the interaction and threshold reset. This level is controlled by the applied current and the leakage parameters. Such synchrony is contingent on significant phase response (that would result, for example, by a sharp PRC jump) occurring during the synaptic rising phase. The rising phase, however, does not contribute significantly if it occurs before the voltage spike reaches its peak. Then a stable near-synchronous state can still exist between type 1 PRC neurons if the PRC shows a left skewness in its shape. These results are examined comprehensively using perfect integrate-and-fire, leaky integrate-and-fire, and skewed PRC shapes under the assumption of the weakly coupled oscillator theory applied to synaptically coupled neuron models.

Qualitative changes in phase-response curve and synchronization at the saddle-node-loop bifurcation

Prominent changes in neuronal dynamics have previously been attributed to a specific switch in onset bifurcation, the Bogdanov-Takens (BT) point. This study unveils another, relevant and so far underestimated transition point: the saddle-node-loop bifurcation, which can be reached by several parameters, including capacitance, leak conductance, and temperature. This bifurcation turns out to induce even more drastic changes in synchronization than the BT transition. This result arises from a direct effect of the saddle-node-loop bifurcation on the limit cycle and hence spike dynamics. In contrast, the BT bifurcation exerts its immediate influence upon the subthreshold dynamics and hence only indirectly relates to spiking. We specifically demonstrate that the saddle-node-loop bifurcation (i) ubiquitously occurs in planar neuron models with a saddle node on invariant cycle onset bifurcation, and (ii) results in a symmetry breaking of the system's phase-response curve. The latter entails an increase in synchronization range in pulse-coupled oscillators, such as neurons. The derived bifurcation structure is of interest in any system for which a relaxation limit is admissible, such as Josephson junctions and chemical oscillators.

Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience

The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear-for example, heteroclinic network attractors. In this review we present a set of mathematical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical framework for further successful applications of mathematics to understanding network dynamics in neuroscience.

Transitions between dynamical behaviors of oscillator networks induced by diversity of nodes and edges

We study the impact of dynamical and structural heterogeneity on the collective dynamics of large small-world networks of pulse-coupled integrate-and-fire oscillators endowed with refractory periods and time delay. Depending on the choice of homogeneous control parameters (here, refractoriness and coupling strength), these networks exhibit a large spectrum of dynamical behaviors, including asynchronous, partially synchronous, and fully synchronous states. Networks exhibit transitions between these dynamical behaviors upon introducing heterogeneity. We show that the probability for a network to exhibit a certain dynamical behavior (network susceptibility) is affected differently by dynamical and structural heterogeneity and depends on the respective homogeneous dynamics.

Changes of Firing Rate Induced by Changes of Phase Response Curve in Bifurcation Transitions

We study dynamical mechanisms responsible for changes of the firing rate during four different bifurcation transitions in the two-dimensional Hindmarsh-Rose (2DHR) neuron model: the saddle node on an invariant circle (SNIC) bifurcation to the supercritical Andronov-Hopf (AH) one, the SNIC bifurcation to the saddle-separatrix loop (SSL) one, the AH bifurcation to the subcritical AH (SAH) one, and the SSL bifurcation to the AH one. For this purpose, we study slopes of the firing rate curve with respect to not only an external input current but also temperature that can be interpreted as a timescale in the 2DHR neuron model. These slopes are mathematically formulated with phase response curves (PRCs), expanding the firing rate with perturbations of the temperature and external input current on the one-dimensional space of the phase [Formula: see text] in the 2DHR oscillator. By analyzing the two different slopes of the firing rate curve with respect to the temperature and external input current, we find that during changes of the firing rate in all of the bifurcation transitions, the calculated slope with respect to the temperature also changes. This is largely dependent on changes in the PRC size that is also related to the slope with respect to the external input current. Furthermore, we find phase transition–like switches of the firing rate with a possible increase of the temperature during the SSL-to-AH bifurcation transition.

Hodge-Kodaira decomposition of evolving neural networks

Although it is very important to scrutinize recurrent structures of neural networks for elucidating brain functions, conventional methods often have difficulty in characterizing global loops within a network systematically. Here we applied the Hodge-Kodaira decomposition, a topological method, to an evolving neural network model in order to characterize its loop structure. By controlling a learning rule parametrically, we found that a model with an STDP-rule, which tends to form paths coincident with causal firing orders, had the most loops. Furthermore, by counting the number of global loops in the network, we detected the inhomogeneity inside the chaotic region, which is usually considered intractable.

Impact of neuronal heterogeneity on correlated colored noise-induced synchronization

Synchronization plays an important role in neural signal processing and transmission. Many hypotheses have been proposed to explain the origin of neural synchronization. In recent years, correlated noise-induced synchronization has received support from many theoretical and experimental studies. However, many of these prior studies have assumed that neurons have identical biophysical properties and that their inputs are well modeled by white noise. In this context, we use colored noise to induce synchronization between oscillators with heterogeneity in both phase-response curves and frequencies. In the low noise limit, we derive novel analytical theory showing that the time constant of colored noise influences correlated noise-induced synchronization and that oscillator heterogeneity can limit synchronization. Surprisingly, however, heterogeneous oscillators may synchronize better than homogeneous oscillators given low input correlations. We also find resonance of oscillator synchronization to colored noise inputs when firing frequencies diverge. Collectively, these results prove robust for both relatively high noise regimes and when applied to biophysically realistic spiking neuron models, and further match experimental recordings from acute brain slices.

Effect of phase response curve skewness on synchronization of electrically coupled neuronal oscillators

We investigate why electrically coupled neuronal oscillators synchronize or fail to synchronize using the theory of weakly coupled oscillators. Stability of synchrony and antisynchrony is predicted analytically and verified using numerical bifurcation diagrams. The shape of the phase response curve (PRC), the shape of the voltage time course, and the frequency of spiking are freely varied to map out regions of parameter spaces that hold stable solutions. We find that type 1 and type 2 PRCs can hold both synchronous and antisynchronous solutions, but the shape of the PRC and the voltage determine the extent of their stability. This is achieved by introducing a five-piecewise linear model to the PRC and a three-piecewise linear model to the voltage time course, and then analyzing the resultant eigenvalue equations that determine the stability of the phase-locked solutions. A single time parameter defines the skewness of the PRC, and another single time parameter defines the spike width and frequency. Our approach gives a comprehensive picture of the relation of the PRC shape, voltage time course, and stability of the resultant synchronous and antisynchronous solutions.

Neural dynamics and information representation in microcircuits of motor cortex

The brain has to analyze and respond to external events that can change rapidly from time to time, suggesting that information processing by the brain may be essentially dynamic rather than static. The dynamical features of neural computation are of significant importance in motor cortex that governs the process of movement generation and learning. In this paper, we discuss these features based primarily on our recent findings on neural dynamics and information coding in the microcircuit of rat motor cortex. In fact, cortical neurons show a variety of dynamical behavior from rhythmic activity in various frequency bands to highly irregular spike firing. Of particular interest are the similarity and dissimilarity of the neuronal response properties in different layers of motor cortex. By conducting electrophysiological recordings in slice preparation, we report the phase response curves (PRCs) of neurons in different cortical layers to demonstrate their layer-dependent synchronization properties. We then study how motor cortex recruits task-related neurons in different layers for voluntary arm movements by simultaneous juxtacellular and multiunit recordings from behaving rats. The results suggest an interesting difference in the spectrum of functional activity between the superficial and deep layers. Furthermore, the task-related activities recorded from various layers exhibited power law distributions of inter-spike intervals (ISIs), in contrast to a general belief that ISIs obey Poisson or Gamma distributions in cortical neurons. We present a theoretical argument that this power law of in vivo neurons may represent the maximization of the entropy of firing rate with limited energy consumption of spike generation. Though further studies are required to fully clarify the functional implications of this coding principle, it may shed new light on information representations by neurons and circuits in motor cortex.

Higher-order spike triggered analysis of neural oscillators

For the purpose of elucidating the neural coding process based on the neural excitability mechanism, researchers have recently investigated the relationship between neural dynamics and the spike triggered stimulus ensemble (STE). Ermentrout et al. analytically derived the relational equation between the phase response curve (PRC) and the spike triggered average (STA). The STA is the first cumulant of the STE. However, in order to understand the neural function as the encoder more explicitly, it is necessary to elucidate the relationship between the PRC and higher-order cumulants of the STE. In this paper, we give a general formulation to relate the PRC and the nth moment of the STE. By using this formulation, we derive a relational equation between the PRC and the spike triggered covariance (STC), which is the covariance of the STE. We show the effectiveness of the relational equation through numerical simulations and use the equation to identify the feature space of the rat hippocampal CA1 pyramidal neurons from their PRCs. Our result suggests that the hippocampal CA1 pyramidal neurons oscillating in the theta frequency range are commonly sensitive to inputs composed of theta and gamma frequency components.

Temperature-modulated synchronization transition in coupled neuronal oscillators

We study two firing properties to characterize the activities of a neuron: frequency-current (f-I) curves and phase response curves (PRCs), with variation in the intrinsic temperature scaling parameter (μ) controlling the opening and closing of ionic channels. We show a peak of the firing frequency for small μ in a class I neuron with the I value immediately after the saddle-node bifurcation, which is entirely different from previous experimental reports as well as model studies. The PRC takes a type II form on a logarithmic f-I curve when μ is small. Then, we analyze the synchronization phenomena in a two-neuron network using the phase-reduction method. We find common μ-dependent transition and bifurcation of synchronizations, regardless of the values of I. Such results give us helpful insight into synchronizations tuned with a sinusoidal-wave temperature modulation on neurons.

Measurement of infinitesimal phase response curves from noisy real neurons

We sought to measure infinitesimal phase response curves (iPRCs) from rat hippocampal CA1 pyramidal neurons. It is difficult to measure iPRCs from noisy neurons because of the dilemma that either the linearity or the signal-to-noise ratio of responses to external perturbations must be sacrificed. To overcome this difficulty, we used an iPRC measurement model formulated as the Langevin phase equation (LPE) to extract iPRCs in the Bayesian scheme. We then simultaneously verified the effectiveness of the measurement model and the reliability of the estimated iPRCs by demonstrating that LPEs with the estimated iPRCs could predict the stochastic behaviors of the same neurons, whose iPRCs had been measured, when they were perturbed by periodic stimulus currents. Our results suggest that the LPE is an effective model for real oscillating neurons and that many theoretical frameworks based on it may be applicable to real nerve systems.

Three synaptic components contributing to robust network synchronization

Robust synchronized activity is widely observed in real neural systems. However, a mechanism for robust synchronization that can be understood analytically, and has a clear physical meaning, remains elusive. This paper considers such a mechanism by formalizing three synaptic components contributing to robust synchronization in networks with heterogeneous external drive currents and conductance-based synapses. The first component arises from the assumption that the aggregate post-synaptic potential of a neuron decays more if it fires later within a spike volley. The second component results because neurons with smaller drives have reached a lower membrane potential at the time when the volley of inputs arrives than that reached by neurons with larger drives. The third component arises from the assumption that neurons firing later in the previous volley have had less time to integrate their drives than neurons firing earlier have had, again causing a lower membrane potential at the time when the volley of inputs arrives. Because of the voltage-dependent properties of synaptic currents, either of the last two components will cause smaller inhibitions for the later-firing neurons if the synapses are inhibitory. This smaller inhibition causes the later-firing neurons to fire earlier in the next cycle, thereby forcing them toward synchrony. With these three synaptic components, we discuss the relationship between the robustness of the synchrony and the parameters, search for the optimal parameter set for the robust network synchronization of a certain frequency band, and demonstrate the key role of the voltage-dependent properties of synaptic currents in robust or stable synchronization.

Phase synchronization between collective rhythms of globally coupled oscillator groups: noiseless nonidentical case

We theoretically study the synchronization between collective oscillations exhibited by two weakly interacting groups of nonidentical phase oscillators with internal and external global sinusoidal couplings of the groups. Coupled amplitude equations describing the collective oscillations of the oscillator groups are obtained by using the Ott-Antonsen ansatz, and then coupled phase equations for the collective oscillations are derived by phase reduction of the amplitude equations. The collective phase coupling function, which determines the dynamics of macroscopic phase differences between the groups, is calculated analytically. We demonstrate that the groups can exhibit effective antiphase collective synchronization even if the microscopic external coupling between individual oscillator pairs belonging to different groups is in-phase, and similarly effective in-phase collective synchronization in spite of microscopic antiphase external coupling between the groups.

Renormalization of oscillator lattices with disorder

A real-space renormalization transformation is constructed for lattices of nonidentical oscillators with dynamics of the general form dvarphi_{k}/dt=omega_{k}+g summation operator_{l}f_{lk}(varphi_{l},varphi_{k}) . The transformation acts on ensembles of such lattices. Critical properties corresponding to a second-order phase transition toward macroscopic synchronization are deduced. The analysis is potentially exact but relies in part on unproven assumptions. Numerically, second-order phase transitions with the predicted properties are observed as g increases in two structurally different two-dimensional oscillator models. One model has smooth coupling f_{lk}(varphi_{l},varphi_{k})=phi(varphi_{l}-varphi_{k}) , where phi(x) is nonodd. The other model is pulse coupled, with f_{lk}(varphi_{l},varphi_{k})=delta(varphi_{l})phi(varphi_{k}) . Lower bounds for the critical dimensions for different types of coupling are obtained. For nonodd coupling, macroscopic synchronization cannot be ruled out for any dimension D> or =1 , whereas in the case of odd coupling, the well-known result that it can be ruled out for D<3 is regained.

Self-organization of a neural network with heterogeneous neurons enhances coherence and stochastic resonance

Most network models for neural behavior assume a predefined network topology and consist of almost identical elements exhibiting little heterogeneity. In this paper, we propose a self-organized network consisting of heterogeneous neurons with different behaviors or degrees of excitability. The synaptic connections evolve according to the spike-timing dependent plasticity mechanism and finally a sparse and active-neuron-dominant structure is observed. That is, strong connections are mainly distributed to the synapses from active neurons to inactive ones. We argue that this self-emergent topology essentially reflects the competition of different neurons and encodes the heterogeneity. This structure is shown to significantly enhance the coherence resonance and stochastic resonance of the entire network, indicating its high efficiency in information processing.

Asynchronous response of coupled pacemaker neurons

We study a network model of two conductance-based pacemaker neurons of differing natural frequency, coupled with either mutual excitation or inhibition, and receiving shared random inhibitory synaptic input. The networks may phase lock spike to spike for strong mutual coupling. But the shared input can desynchronize the locked spike pairs by selectively eliminating the lagging spike or modulating its timing with respect to the leading spike depending on their separation time window. Such loss of synchrony is also found in a large network of sparsely coupled heterogeneous spiking neurons receiving shared input.

Complex evolution of spike patterns during burst propagation through feed-forward networks

Stable signal transmission is crucial for information processing by the brain. Synfire-chains, defined as feed-forward networks of spiking neurons, are a well-studied class of circuit structure that can propagate a packet of single spikes while maintaining a fixed packet profile. Here, we studied the stable propagation of spike bursts, rather than single spike activities, in a feed-forward network of a general class of excitable bursting neurons. In contrast to single spikes, bursts can propagate stably without converging to any fixed profiles. Spike timings of bursts continue to change cyclically or irregularly during propagation depending on intrinsic properties of the neurons and the coupling strength of the network. To find the conditions under which bursts lose fixed profiles, we propose an analysis based on timing shifts of burst spikes similar to the phase response analysis of limit-cycle oscillators.