Phase-resetting curves determine synchronization, phase locking, and clustering in networks of neural oscillators

Phase-resetting as a tool of information transmission

Models of information transmission in the brain largely rely on firing rate codes. The abundance of oscillatory activity in the brain suggests that information may be also encoded using the phases of ongoing oscillations. Sensory perception, working memory and spatial navigation have been hypothesized to use phase codes, and cross-frequency coordination and phase synchronization between brain areas have been proposed to gate the flow of information. Phase codes generally require the phase of the oscillations to be reset at specific reference points for consistent coding, and coordination between oscillators requires favorable phase resetting characteristics. Recent evidence supports a role for neural oscillations in providing temporal reference windows that allow for correct parsing of phase-coded information.

Zero-lag synchronization despite inhomogeneities in a relay system

A novel proposal for the zero-lag synchronization of the delayed coupled neurons, is to connect them indirectly via a third relay neuron. In this study, we develop a Poincaré map to investigate the robustness of the synchrony in such a relay system against inhomogeneity in the neurons and synaptic parameters. We show that when the inhomogeneity does not violate the symmetry of the system, synchrony is maintained and in some cases inhomogeneity enhances synchrony. On the other hand if the inhomogeneity breaks the symmetry of the system, zero lag synchrony can not be preserved. In this case we give analytical results for the phase lag of the spiking of the neurons in the stable state.

Effect of autaptic activity on the response of a Hodgkin-Huxley neuron

An autapse is a special synapse that connects a neuron to itself. In this study, we investigated the effect of an autapse on the responses of a Hodgkin-Huxley neuron to different forms of external stimuli. When the neuron was subjected to a DC stimulus, the firing frequencies and the interspike interval distributions of the output spike trains showed periodic behaviors as the autaptic delay time increased. When the input was a synaptic pulse-like train with random interspike intervals, we observed low-pass and band-pass filtering behaviors. Moreover, the region over which the output ISIs are distributed and the mean firing frequency display periodic behaviors with increasing autaptic delay time. When specific autaptic parameters were chosen, most of the input ISIs could be filtered, and the response spike trains were nearly regular, even with a highly random input. The background mechanism of these observed dynamics has been analyzed based on the phase response curve method. We also found that the information entropy of the output spike train could be modified by the autapse. These results also suggest that the autapse can serve as a regulator of information response in the nervous system.

Predicting the responses of repetitively firing neurons to current noise

We used phase resetting methods to predict firing patterns of rat subthalamic nucleus (STN) neurons when their rhythmic firing was densely perturbed by noise. We applied sequences of contiguous brief (0.5–2 ms) current pulses with amplitudes drawn from a Gaussian distribution (10–100 pA standard deviation) to autonomously firing STN neurons in slices. Current noise sequences increased the variability of spike times with little or no effect on the average firing rate. We measured the infinitesimal phase resetting curve (PRC) for each neuron using a noise-based method. A phase model consisting of only a firing rate and PRC was very accurate at predicting spike timing, accounting for more than 80% of spike time variance and reliably reproducing the spike-to-spike pattern of irregular firing. An approximation for the evolution of phase was used to predict the effect of firing rate and noise parameters on spike timing variability. It quantitatively predicted changes in variability of interspike intervals with variation in noise amplitude, pulse duration and firing rate over the normal range of STN spontaneous rates. When constant current was used to drive the cells to higher rates, the PRC was altered in size and shape and accurate predictions of the effects of noise relied on incorporating these changes into the prediction. Application of rate-neutral changes in conductance showed that changes in PRC shape arise from conductance changes known to accompany rate increases in STN neurons, rather than the rate increases themselves. Our results show that firing patterns of densely perturbed oscillators cannot readily be distinguished from those of neurons randomly excited to fire from the rest state. The spike timing of repetitively firing neurons may be quantitatively predicted from the input and their PRCs, even when they are so densely perturbed that they no longer fire rhythmically.

Most neurons receive thousands of synaptic inputs per second. Each of these may be individually weak but collectively they shape the temporal pattern of firing by the postsynaptic neuron. If the postsynaptic neuron fires repetitively, its synaptic inputs need not directly trigger action potentials, but may instead control the timing of action potentials that would occur anyway. The phase resetting curve encapsulates the influence of an input on the timing of the next action potential, depending on its time of arrival. We measured the phase resetting curves of neurons in the subthalamic nucleus and used them to accurately predict the timing of action potentials in a phase model subjected to complex input patterns. A simple approximation to the phase model accurately predicted the changes in firing pattern evoked by dense patterns of noise pulses varying in amplitude and pulse duration, and by changes in firing rate. We also showed that the phase resetting curve changes systematically with changes in total neuron conductance, and doing so predicts corresponding changes in firing pattern. Our results indicate that the phase model may accurately represent the temporal integration of complex patterns of input to repetitively firing neurons.

Interplay of intrinsic and synaptic conductances in the generation of high-frequency oscillations in interneuronal networks with irregular spiking

High-frequency oscillations (above 30 Hz) have been observed in sensory and higher-order brain areas, and are believed to constitute a general hallmark of functional neuronal activation. Fast inhibition in interneuronal networks has been suggested as a general mechanism for the generation of high-frequency oscillations. Certain classes of interneurons exhibit subthreshold oscillations, but the effect of this intrinsic neuronal property on the population rhythm is not completely understood. We study the influence of intrinsic damped subthreshold oscillations in the emergence of collective high-frequency oscillations, and elucidate the dynamical mechanisms that underlie this phenomenon. We simulate neuronal networks composed of either Integrate-and-Fire (IF) or Generalized Integrate-and-Fire (GIF) neurons. The IF model displays purely passive subthreshold dynamics, while the GIF model exhibits subthreshold damped oscillations. Individual neurons receive inhibitory synaptic currents mediated by spiking activity in their neighbors as well as noisy synaptic bombardment, and fire irregularly at a lower rate than population frequency. We identify three factors that affect the influence of single-neuron properties on synchronization mediated by inhibition: i) the firing rate response to the noisy background input, ii) the membrane potential distribution, and iii) the shape of Inhibitory Post-Synaptic Potentials (IPSPs). For hyperpolarizing inhibition, the GIF IPSP profile (factor iii)) exhibits post-inhibitory rebound, which induces a coherent spike-mediated depolarization across cells that greatly facilitates synchronous oscillations. This effect dominates the network dynamics, hence GIF networks display stronger oscillations than IF networks. However, the restorative current in the GIF neuron lowers firing rates and narrows the membrane potential distribution (factors i) and ii), respectively), which tend to decrease synchrony. If inhibition is shunting instead of hyperpolarizing, post-inhibitory rebound is not elicited and factors i) and ii) dominate, yielding lower synchrony in GIF networks than in IF networks.

Neurons in the brain engage in collective oscillations at different frequencies. Gamma and high-gamma oscillations (30–100 Hz and higher) have been associated with cognitive functions, and are altered in psychiatric disorders such as schizophrenia and autism. Our understanding of how high-frequency oscillations are orchestrated in the brain is still limited, but it is necessary for the development of effective clinical approaches to the treatment of these disorders. Some neuron types exhibit dynamical properties that can favour synchronization. The theory of weakly coupled oscillators showed how the phase response of individual neurons can predict the patterns of phase relationships that are observed at the network level. However, neurons in vivo do not behave like regular oscillators, but fire irregularly in a regime dominated by fluctuations. Hence, which intrinsic dynamical properties matter for synchronization, and in which regime, is still an open question. Here, we show how single-cell damped subthreshold oscillations enhance synchrony in interneuronal networks by introducing a depolarizing component, mediated by post-inhibitory rebound, that is correlated among neurons due to common inhibitory input.

Slow noise in the period of a biological oscillator underlies gradual trends and abrupt transitions in phasic relationships in hybrid neural networks

In order to study the ability of coupled neural oscillators to synchronize in the presence of intrinsic as opposed to synaptic noise, we constructed hybrid circuits consisting of one biological and one computational model neuron with reciprocal synaptic inhibition using the dynamic clamp. Uncoupled, both neurons fired periodic trains of action potentials. Most coupled circuits exhibited qualitative changes between one-to-one phase-locking with fairly constant phasic relationships and phase slipping with a constant progression in the phasic relationships across cycles. The phase resetting curve (PRC) and intrinsic periods were measured for both neurons, and used to construct a map of the firing intervals for both the coupled and externally forced (PRC measurement) conditions. For the coupled network, a stable fixed point of the map predicted phase locking, and its absence produced phase slipping. Repetitive application of the map was used to calibrate different noise models to simultaneously fit the noise level in the measurement of the PRC and the dynamics of the hybrid circuit experiments. Only a noise model that added history-dependent variability to the intrinsic period could fit both data sets with the same parameter values, as well as capture bifurcations in the fixed points of the map that cause switching between slipping and locking. We conclude that the biological neurons in our study have slowly-fluctuating stochastic dynamics that confer history dependence on the period. Theoretical results to date on the behavior of ensembles of noisy biological oscillators may require re-evaluation to account for transitions induced by slow noise dynamics.

Effect of heterogeneity and noise on cross frequency phase-phase and phase-amplitude coupling

Cross-frequency coupling is hypothesized to play a functional role in neural computation. We apply phase resetting theory to two types of cross-frequency coupling that can occur when a slower oscillator periodically forces one or more oscillators: phase-phase coupling, in which the two oscillations are phase-locked, and phase-amplitude coupling, in which the amplitude of the driven oscillation is modulated. Our first result is that the shape of the phase resetting curve predicts the tightness of locking to a pulsatile forcing periodic input at any ratio of forced to intrinsic period; the tightness of the locking decreases as the ratio increases. Theoretical expressions were obtained for the probability density of the phases for a population of heterogeneous oscillators or a noisy single oscillator. Results were confirmed using two types of simulated networks and experiments on hippocampal CA1 neurons. Theoretical expressions were also obtained and confirmed for the probability density of N spike times within a single cycle of low frequency forcing. The second result is a suggested mechanism for phase-amplitude coupling in which progressive desynchronization leads to decreasing amplitude during a low frequency forcing cycle. Network simulations confirmed the theoretical viability of this mechanism, and that it generalizes to more diffuse input.

Effects of synaptic plasticity on phase and period locking in a network of two oscillatory neurons

We study the effects of synaptic plasticity on the determination of firing period and relative phases in a network of two oscillatory neurons coupled with reciprocal inhibition. We combine the phase response curves of the neurons with the short-term synaptic plasticity properties of the synapses to define Poincaré maps for the activity of an oscillatory network. Fixed points of these maps correspond to the phase-locked modes of the network. These maps allow us to analyze the dependence of the resulting network activity on the properties of network components. Using a combination of analysis and simulations, we show how various parameters of the model affect the existence and stability of phase-locked solutions. We find conditions on the synaptic plasticity profiles and the phase response curves of the neurons for the network to be able to maintain a constant firing period, while varying the phase of locking between the neurons or vice versa. A generalization to cobwebbing for two-dimensional maps is also discussed.

Effect of phase response curve skew on synchronization with and without conduction delays

A central problem in cortical processing including sensory binding and attentional gating is how neurons can synchronize their responses with zero or near-zero time lag. For a spontaneously firing neuron, an input from another neuron can delay or advance the next spike by different amounts depending upon the timing of the input relative to the previous spike. This information constitutes the phase response curve (PRC). We present a simple graphical method for determining the effect of PRC shape on synchronization tendencies and illustrate it using type 1 PRCs, which consist entirely of advances (delays) in response to excitation (inhibition). We obtained the following generic solutions for type 1 PRCs, which include the pulse-coupled leaky integrate and fire model. For pairs with mutual excitation, exact synchrony can be stable for strong coupling because of the stabilizing effect of the causal limit region of the PRC in which an input triggers a spike immediately upon arrival. However, synchrony is unstable for short delays, because delayed inputs arrive during a refractory period and cannot trigger an immediate spike. Right skew destabilizes antiphase and enables modes with time lags that grow as the conduction delay is increased. Therefore, right skew favors near synchrony at short conduction delays and a gradual transition between synchrony and antiphase for pairs coupled by mutual excitation. For pairs with mutual inhibition, zero time lag synchrony is stable for conduction delays ranging from zero to a substantial fraction of the period for pairs. However, for right skew there is a preferred antiphase mode at short delays. In contrast to mutual excitation, left skew destabilizes antiphase for mutual inhibition so that synchrony dominates at short delays as well. These pairwise synchronization tendencies constrain the synchronization properties of neurons embedded in larger networks.

Multipulse phase resetting curves

In this paper, we introduce and study systematically, in terms of phase response curves, the effect of dual-pulse excitation on the dynamics of an autonomous oscillator. Specifically, we test the deviations from linear summation of phase advances resulting from two small perturbations. We analytically derive a correction term, which generally appears for oscillators whose intrinsic dimensionality is >1. The nonlinear correction term is found to be proportional to the square of the perturbation. We demonstrate this effect in the Stuart-Landau model and in various higher dimensional neuronal models. This deviation from the superposition principle needs to be taken into account in studies of networks of pulse-coupled oscillators. Further, this deviation could be used in the verification of oscillator models via a dual-pulse excitation.

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.

Phase description of stochastic oscillations

We introduce an invariant phase description of stochastic oscillations by generalizing the concept of standard isophases. The average isophases are constructed as sections in the state space, having a constant mean first return time. The approach allows us to obtain a global phase variable of noisy oscillations, even in the cases where the phase is ill defined in the deterministic limit. A simple numerical method for finding the isophases is illustrated for noise-induced switching between two coexisting limit cycles, and for noise-induced oscillation in an excitable system. We also discuss how to determine isophases of observed irregular oscillations, providing a basis for a refined phase description in data analysis.

Toggling between gamma-frequency activity and suppression of cell assemblies

Gamma (30–80 Hz) rhythms in hippocampus and neocortex resulting from the interaction of excitatory and inhibitory cells (E- and I-cells), called Pyramidal-Interneuronal Network Gamma (PING), require that the I-cells respond to the E-cells, but don't fire on their own. In idealized models, there is a sharp boundary between a parameter regime where the I-cells have weak-enough drive for PING, and one where they have so much drive that they fire without being prompted by the E-cells. In the latter regime, they often de-synchronize and suppress the E-cells; the boundary was therefore called the “suppression boundary” by Börgers and Kopell (2005). The model I-cells used in the earlier work by Börgers and Kopell have a “type 1” phase response, i.e., excitatory input always advances them. However, fast-spiking inhibitory basket cells often have a “type 2” phase response: Excitatory input arriving soon after they fire delays them. We study the effect of the phase response type on the suppression transition, under the additional assumption that the I-cells are kept synchronous by gap junctions. When many E-cells participate on a given cycle, the resulting excitation advances the I-cells on the next cycle if their phase response is of type 1, and this can result in suppression of more E-cells on the next cycle. Therefore, strong E-cell spike volleys tend to be followed by weaker ones, and vice versa. This often results in erratic fluctuations in the strengths of the E-cell spike volleys. When the phase response of the I-cells is of type 2, the opposite happens: strong E-cell spike volleys delay the inhibition on the next cycle, therefore tend to be followed by yet stronger ones. The strengths of the E-cell spike volleys don't oscillate, and there is a nearly abrupt transition from PING to ING (a rhythm involving I-cells only).

Effect of sharp jumps at the edges of phase response curves on synchronization of electrically coupled neuronal oscillators

We study synchronization phenomenon of coupled neuronal oscillators using the theory of weakly coupled oscillators. The role of sudden jumps in the phase response curve profiles found in some experimental recordings and models on the ability of coupled neurons to exhibit synchronous and antisynchronous behavior is investigated, when the coupling between the neurons is electrical. The level of jumps in the phase response curve at either end, spike width and frequency of voltage time course of the coupled neurons are parameterized using piecewise linear functional forms, and the conditions for stable synchrony and stable antisynchrony in terms of those parameters are computed analytically. The role of the peak position of the phase response curve on phase-locking is also investigated.

Parameterized phase response curves for characterizing neuronal behaviors under transient conditions

Phase response curves (PRCs) are a simple model of how a neuron's spike time is affected by synaptic inputs. PRCs are useful in predicting how networks of neurons behave when connected. One challenge in estimating a neuron's PRCs experimentally is that many neurons do not have stationary firing rates. In this article we introduce a new method to estimate PRCs as a function of firing rate of the neuron. We call the resulting model a parameterized PRC (pPRC). Experimentally, we perturb the neuron applying a current with two parts: 1) a current held constant between spikes but changed at the onset of a spike, used to make the neuron fire at different rates, and 2) a pulse to emulate a synaptic input. A model of the applied constant current and the history is made to predict the interspike interval (ISI). A second model is then made to fit the modulation of the spike time from the expected ISI by the pulsatile stimulus. A polynomial with two independent variables, the stimulus phase and the expected ISI, is used to model the pPRC. The pPRC is validated in a computational model and applied to pyramidal neurons from the CA1 region of the hippocampal slices from rat. The pPRC can be used to model the effect of changing firing rates on network synchrony. It can also be used to characterize the effects of neuromodulators and genetic mutations (among other manipulations) on network synchrony. It can also easily be extended to account for more variables.

Phase-amplitude descriptions of neural oscillator models

Phase oscillators are a common starting point for the reduced description of many single neuron models that exhibit a strongly attracting limit cycle. The framework for analysing such models in response to weak perturbations is now particularly well advanced, and has allowed for the development of a theory of weakly connected neural networks. However, the strong-attraction assumption may well not be the natural one for many neural oscillator models. For example, the popular conductance based Morris-Lecar model is known to respond to periodic pulsatile stimulation in a chaotic fashion that cannot be adequately described with a phase reduction. In this paper, we generalise the phase description that allows one to track the evolution of distance from the cycle as well as phase on cycle. We use a classical technique from the theory of ordinary differential equations that makes use of a moving coordinate system to analyse periodic orbits. The subsequent phase-amplitude description is shown to be very well suited to understanding the response of the oscillator to external stimuli (which are not necessarily weak). We consider a number of examples of neural oscillator models, ranging from planar through to high dimensional models, to illustrate the effectiveness of this approach in providing an improvement over the standard phase-reduction technique. As an explicit application of this phase-amplitude framework, we consider in some detail the response of a generic planar model where the strong-attraction assumption does not hold, and examine the response of the system to periodic pulsatile forcing. In addition, we explore how the presence of dynamical shear can lead to a chaotic response.

Noxious somatic stimuli diminish respiratory-sympathetic coupling by selective resetting of the respiratory rhythm in anaesthetized rats

Noxious somatic stimulation evokes respiratory and autonomic responses. The mechanisms underlying the responses and the manner in which they are co-ordinated are still unclear. The effects of activation of somatic nociceptive fibres on lumbar sympathetic nerve activity at slow (2-10 Hz) and fast frequency bands (100-1000 Hz) and the effects on respiratory-sympathetic coupling are unknown. In anaesthetized, artificially ventilated Sprague-Dawley rats under neuromuscular blockade, ensemble averaging of sympathetic activity following high-intensity single-pulse stimulation of the sciatic nerve revealed two peaks (~140 and ~250 ms) that were present at similar latencies whether or not slow or fast band filtering was used. Additionally, in the slow band of both lumbar and splanchnic sympathetic nerve activity, a third peak with a very slow latency (~650 ms) was apparent. In the respiratory system, activation of the sciatic nerve decreased the expiratory period when the stimulus occurred during the first half of expiration, but increased the expiratory period if the stimulus was delivered in the second half of the expiratory phase. The phase shifting of the respiratory cycle also impaired the respiratory-sympathetic coupling in both splanchnic and lumbar sympathetic nerve activity in the subsequent respiratory cycle. The findings suggest that noxious somatosympathetic responses reduce the co-ordination between respiration and perfusion by resetting the respiratory pattern generator.

Short conduction delays cause inhibition rather than excitation to favor synchrony in hybrid neuronal networks of the entorhinal cortex

How stable synchrony in neuronal networks is sustained in the presence of conduction delays is an open question. The Dynamic Clamp was used to measure phase resetting curves (PRCs) for entorhinal cortical cells, and then to construct networks of two such neurons. PRCs were in general Type I (all advances or all delays) or weakly type II with a small region at early phases with the opposite type of resetting. We used previously developed theoretical methods based on PRCs under the assumption of pulsatile coupling to predict the delays that synchronize these hybrid circuits. For excitatory coupling, synchrony was predicted and observed only with no delay and for delays greater than half a network period that cause each neuron to receive an input late in its firing cycle and almost immediately fire an action potential. Synchronization for these long delays was surprisingly tight and robust to the noise and heterogeneity inherent in a biological system. In contrast to excitatory coupling, inhibitory coupling led to antiphase for no delay, very short delays and delays close to a network period, but to near-synchrony for a wide range of relatively short delays. PRC-based methods show that conduction delays can stabilize synchrony in several ways, including neutralizing a discontinuity introduced by strong inhibition, favoring synchrony in the case of noisy bistability, and avoiding an initial destabilizing region of a weakly type II PRC. PRCs can identify optimal conduction delays favoring synchronization at a given frequency, and also predict robustness to noise and heterogeneity.

Individual oscillators, such as pendulum-based clocks and fireflies, can spontaneously organize into a coherent, synchronized entity with a common frequency. Neurons can oscillate under some circumstances, and can synchronize their firing both within and across brain regions. Synchronized assemblies of neurons are thought to underlie cognitive functions such as recognition, recall, perception and attention. Pathological synchrony can lead to epilepsy, tremor and other dynamical diseases, and synchronization is altered in most mental disorders. Biological neurons synchronize despite conduction delays, heterogeneous circuit composition, and noise. In biological experiments, we built simple networks in which two living neurons could interact via a computer in real time. The computer precisely controlled the nature of the connectivity and the length of the communication delays. We characterized the synchronization tendencies of individual, isolated oscillators by measuring how much a single input delivered by the computer transiently shortened or lengthened the cycle period of the oscillation. We then used this information to correctly predict the strong dependence of the coordination pattern of the firing of the component neurons on the length of the communication delays. Upon this foundation, we can begin to build a theory of the basic principles of synchronization in more complex brain circuits.

Dynamical changes in neurons during seizures determine tonic to clonic shift

A tonic-clonic seizure transitions from high frequency asynchronous activity to low frequency coherent oscillations, yet the mechanism of transition remains unknown. We propose a shift in network synchrony due to changes in cellular response. Here we use phase-response curves (PRC) from Morris-Lecar (M-L) model neurons with synaptic depression and gradually decrease input current to cells within a network simulation. This method effectively decreases firing rates resulting in a shift to greater network synchrony illustrating a possible mechanism of the transition phenomenon. PRCs are measured from the M-L conductance based model cell with a range of input currents within the limit cycle. A large network of 3000 excitatory neurons is simulated with a network topology generated from second-order statistics which allows a range of population synchrony. The population synchrony of the oscillating cells is measured with the Kuramoto order parameter, which reveals a transition from tonic to clonic phase exhibited by our model network. The cellular response shift mechanism for the tonic-clonic seizure transition reproduces the population behavior closely when compared to EEG data.

Effects of conduction delays on the existence and stability of one to one phase locking between two pulse-coupled oscillators

Gamma oscillations can synchronize with near zero phase lag over multiple cortical regions and between hemispheres, and between two distal sites in hippocampal slices. How synchronization can take place over long distances in a stable manner is considered an open question. The phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike, depending upon where in the cycle it is received. We use PRCs under the assumption of pulsatile coupling to derive existence and stability criteria for 1:1 phase-locking that arises via bidirectional pulse coupling of two limit cycle oscillators with a conduction delay of any duration for any 1:1 firing pattern. The coupling can be strong as long as the effect of one input dissipates before the next input is received. We show the form that the generic synchronous and anti-phase solutions take in a system of two identical, identically pulse-coupled oscillators with identical delays. The stability criterion has a simple form that depends only on the slopes of the PRCs at the phases at which inputs are received and on the number of cycles required to complete the delayed feedback loop. The number of cycles required to complete the delayed feedback loop depends upon both the value of the delay and the firing pattern. We successfully tested the predictions of our methods on networks of model neurons. The criteria can easily be extended to include the effect of an input on the cycle after the one in which it is received.

Functional connectivity and integrative properties of globus pallidus neurons

The globus pallidus consists of the external (GPe) and the internal (GPi) segments. The GPe and GPi have different functional roles. The GPe is located centrally within multiple basal ganglia feedforward and feedback connections. The GPi is an output nucleus of the basal ganglia. A complex interplay between intrinsic pacemaking conductances and the balance of glutamatergic and GABAergic input largely determines the rate and pattern of firing of pallidal neurons. The initial part of this article introduces recent findings made in vivo that are related to the roles of glutamatergic and GABAergic inputs in the control of pallidal activity. The latter part describes the roles of intrinsic mechanisms of GPe neurons in the integration of the synaptic inputs. The presence of dendritic voltage-gated sodium channels may allow the initiation of dendritic spikes, giving distal inputs on the long and thin GPe dendrites an opportunity to strongly shape spiking activity. Basal ganglia disorders including Parkinson's disease, hemiballismus, and dystonias are accompanied by increased irregularity and synchronized bursts of pallidal activity. These changes may be in part due to changes in the GABA release in the GPe and GPi, but also involve intrinsic cellular changes in pallidal neurons.

Network reconstruction from random phase resetting

We propose a novel method of reconstructing the topology and interaction functions for a general oscillator network. An ensemble of initial phases and the corresponding instantaneous frequencies is constructed by repeating random phase resets of the system dynamics. The desired details of network structure are then revealed by appropriately averaging over the ensemble. The method is applicable for a wide class of networks with arbitrary emergent dynamics, including full synchrony.

Multi-scale modeling in biology: how to bridge the gaps between scales?

Human physiological functions are regulated across many orders of magnitude in space and time. Integrating the information and dynamics from one scale to another is critical for the understanding of human physiology and the treatment of diseases. Multi-scale modeling, as a computational approach, has been widely adopted by researchers in computational and systems biology. A key unsolved issue is how to represent appropriately the dynamical behaviors of a high-dimensional model of a lower scale by a low-dimensional model of a higher scale, so that it can be used to investigate complex dynamical behaviors at even higher scales of integration. In the article, we first review the widely-used different modeling methodologies and their applications at different scales. We then discuss the gaps between different modeling methodologies and between scales, and discuss potential methods for bridging the gaps between scales.

Cellularly-driven differences in network synchronization propensity are differentially modulated by firing frequency

Spatiotemporal pattern formation in neuronal networks depends on the interplay between cellular and network synchronization properties. The neuronal phase response curve (PRC) is an experimentally obtainable measure that characterizes the cellular response to small perturbations, and can serve as an indicator of cellular propensity for synchronization. Two broad classes of PRCs have been identified for neurons: Type I, in which small excitatory perturbations induce only advances in firing, and Type II, in which small excitatory perturbations can induce both advances and delays in firing. Interestingly, neuronal PRCs are usually attenuated with increased spiking frequency, and Type II PRCs typically exhibit a greater attenuation of the phase delay region than of the phase advance region. We found that this phenomenon arises from an interplay between the time constants of active ionic currents and the interspike interval. As a result, excitatory networks consisting of neurons with Type I PRCs responded very differently to frequency modulation compared to excitatory networks composed of neurons with Type II PRCs. Specifically, increased frequency induced a sharp decrease in synchrony of networks of Type II neurons, while frequency increases only minimally affected synchrony in networks of Type I neurons. These results are demonstrated in networks in which both types of neurons were modeled generically with the Morris-Lecar model, as well as in networks consisting of Hodgkin-Huxley-based model cortical pyramidal cells in which simulated effects of acetylcholine changed PRC type. These results are robust to different network structures, synaptic strengths and modes of driving neuronal activity, and they indicate that Type I and Type II excitatory networks may display two distinct modes of processing information.

Synchronization of the firing of neurons in the brain is related to many cognitive functions, such as recognizing faces, discriminating odors, and coordinating movement. It is therefore important to understand what properties of neuronal networks promote synchrony of neural firing. One measure that is often used to determine the contribution of individual neurons to network synchrony is called the phase response curve (PRC). PRCs describe how the timing of neuronal firing changes depending on when input, such as a synaptic signal, is received by the neuron. A characteristic of PRCs that has previously not been well understood is that they change dramatically as the neuron's firing frequency is modulated. This effect carries potential significance, since cognitive functions are often associated with specific frequencies of network activity in the brain. We showed computationally that the frequency dependence of PRCs can be explained by the relative timing of ionic membrane currents with respect to the time between spike firings. Our simulations also showed that the frequency dependence of neuronal PRCs leads to frequency-dependent changes in network synchronization that can be different for different neuron types. These results further our understanding of how synchronization is generated in the brain to support various cognitive functions.

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.

Variability, compensation, and modulation in neurons and circuits

I summarize recent computational and experimental work that addresses the inherent variability in the synaptic and intrinsic conductances in normal healthy brains and shows that multiple solutions (sets of parameters) can produce similar circuit performance. I then discuss a number of issues raised by this observation, such as which parameter variations arise from compensatory mechanisms and which reflect insensitivity to those particular parameters. I ask whether networks with different sets of underlying parameters can nonetheless respond reliably to neuromodulation and other global perturbations. At the computational level, I describe a paradigm shift in which it is becoming increasingly common to develop families of models that reflect the variance in the biological data that the models are intended to illuminate rather than single, highly tuned models. On the experimental side, I discuss the inherent limitations of overreliance on mean data and suggest that it is important to look for compensations and correlations among as many system parameters as possible, and between each system parameter and circuit performance. This second paradigm shift will require moving away from measurements of each system component in isolation but should reveal important previously undescribed principles in the organization of complex systems such as brains.

Synchronous bursts on scale-free neuronal networks with attractive and repulsive coupling

This paper investigates the dependence of synchronization transitions of bursting oscillations on the information transmission delay over scale-free neuronal networks with attractive and repulsive coupling. It is shown that for both types of coupling, the delay always plays a subtle role in either promoting or impairing synchronization. In particular, depending on the inherent oscillation period of individual neurons, regions of irregular and regular propagating excitatory fronts appear intermittently as the delay increases. These delay-induced synchronization transitions are manifested as well-expressed minima in the measure for spatiotemporal synchrony. For attractive coupling, the minima appear at every integer multiple of the average oscillation period, while for the repulsive coupling, they appear at every odd multiple of the half of the average oscillation period. The obtained results are robust to the variations of the dynamics of individual neurons, the system size, and the neuronal firing type. Hence, they can be used to characterize attractively or repulsively coupled scale-free neuronal networks with delays.

The variance of phase-resetting curves

Phase resetting curves (PRCs) provide a measure of the sensitivity of oscillators to perturbations. In a noisy environment, these curves are themselves very noisy. Using perturbation theory, we compute the mean and the variance for PRCs for arbitrary limit cycle oscillators when the noise is small. Phase resetting curves and phase dependent variance are fit to experimental data and the variance is computed using an ad-hoc method. The theoretical curves of this phase dependent method match both simulations and experimental data significantly better than an ad-hoc method. A dual cell network simulation is compared to predictions using the analytical phase dependent variance estimation presented in this paper. We also discuss how entrainment of a neuron to a periodic pulse depends on the noise amplitude.

Non-weak inhibition and phase resetting at negative values of phase in cells with fast-slow dynamics at hyperpolarized potentials

Phase response is a powerful concept in the analysis of both weakly and non-weakly perturbed oscillators such as regularly spiking neurons, and is applicable if the oscillator returns to its limit cycle trajectory between successive perturbations. When the latter condition is violated, a formal application of the phase return map may yield phase values outside of its definition domain; in particular, strong synaptic inhibition may result in negative values of phase. The effect of a second perturbation arriving close to the first one is undetermined in this case. However, here we show that for a Morris-Lecar model of a spiking cell with strong time scale separation, extending the phase response function definition domain to an additional negative value branch allows to retain the accuracy of the phase response approach in the face of such strong inhibitory coupling. We use the resulting extended phase response function to accurately describe the response of a Morris-Lecar oscillator to consecutive non-weak synaptic inputs. This method is particularly useful when analyzing the dynamics of three or more non-weakly coupled cells, whereby more than one synaptic perturbation arrives per oscillation cycle into each cell. The method of perturbation prediction based on the negative-phase extension of the phase response function may be applicable to other excitable cell models characterized by slow voltage dynamics at hyperpolarized potentials.

Phase resetting of collective rhythm in ensembles of oscillators

Phase resetting curves characterize the way a system with a collective periodic behavior responds to perturbations. We consider globally coupled ensembles of Sakaguchi-Kuramoto oscillators, and use the Ott-Antonsen theory of ensemble evolution to derive the analytical phase resetting equations. We show the final phase reset value to be composed of two parts: an immediate phase reset directly caused by the perturbation and the dynamical phase reset resulting from the relaxation of the perturbed system back to its dynamical equilibrium. Analytical, semianalytical and numerical approximations of the final phase resetting curve are constructed. We support our findings with extensive numerical evidence involving identical and nonidentical oscillators. The validity of our theory is discussed in the context of large ensembles approximating the thermodynamic limit.

Phase-response curves and synchronized neural networks

We review the principal assumptions underlying the application of phase-response curves (PRCs) to synchronization in neuronal networks. The PRC measures how much a given synaptic input perturbs spike timing in a neural oscillator. Among other applications, PRCs make explicit predictions about whether a given network of interconnected neurons will synchronize, as is often observed in cortical structures. Regarding the assumptions of the PRC theory, we conclude: (i) The assumption of noise-tolerant cellular oscillations at or near the network frequency holds in some but not all cases. (ii) Reduced models for PRC-based analysis can be formally related to more realistic models. (iii) Spike-rate adaptation limits PRC-based analysis but does not invalidate it. (iv) The dependence of PRCs on synaptic location emphasizes the importance of improving methods of synaptic stimulation. (v) New methods can distinguish between oscillations that derive from mutual connections and those arising from common drive. (vi) It is helpful to assume linear summation of effects of synaptic inputs; experiments with trains of inputs call this assumption into question. (vii) Relatively subtle changes in network structure can invalidate PRC-based predictions. (viii) Heterogeneity in the preferred frequencies of component neurons does not invalidate PRC analysis, but can annihilate synchronous activity.

Stability of two cluster solutions in pulse coupled networks of neural oscillators

Phase response curves (PRCs) have been widely used to study synchronization in neural circuits comprised of pacemaking neurons. They describe how the timing of the next spike in a given spontaneously firing neuron is affected by the phase at which an input from another neuron is received. Here we study two reciprocally coupled clusters of pulse coupled oscillatory neurons. The neurons within each cluster are presumed to be identical and identically pulse coupled, but not necessarily identical to those in the other cluster. We investigate a two cluster solution in which all oscillators are synchronized within each cluster, but in which the two clusters are phase locked at nonzero phase with each other. Intuitively, one might expect this solution to be stable only when synchrony within each isolated cluster is stable, but this is not the case. We prove rigorously the stability of the two cluster solution and show how reciprocal coupling can stabilize synchrony within clusters that cannot synchronize in isolation. These stability results for the two cluster solution suggest a mechanism by which reciprocal coupling between brain regions can induce local synchronization via the network feedback loop.

Neurophysiological and computational principles of cortical rhythms in cognition

Synchronous rhythms represent a core mechanism for sculpting temporal coordination of neural activity in the brain-wide network. This review focuses on oscillations in the cerebral cortex that occur during cognition, in alert behaving conditions. Over the last two decades, experimental and modeling work has made great strides in elucidating the detailed cellular and circuit basis of these rhythms, particularly gamma and theta rhythms. The underlying physiological mechanisms are diverse (ranging from resonance and pacemaker properties of single cells to multiple scenarios for population synchronization and wave propagation), but also exhibit unifying principles. A major conceptual advance was the realization that synaptic inhibition plays a fundamental role in rhythmogenesis, either in an interneuronal network or in a reciprocal excitatory-inhibitory loop. Computational functions of synchronous oscillations in cognition are still a matter of debate among systems neuroscientists, in part because the notion of regular oscillation seems to contradict the common observation that spiking discharges of individual neurons in the cortex are highly stochastic and far from being clocklike. However, recent findings have led to a framework that goes beyond the conventional theory of coupled oscillators and reconciles the apparent dichotomy between irregular single neuron activity and field potential oscillations. From this perspective, a plethora of studies will be reviewed on the involvement of long-distance neuronal coherence in cognitive functions such as multisensory integration, working memory, and selective attention. Finally, implications of abnormal neural synchronization are discussed as they relate to mental disorders like schizophrenia and autism.

Synaptic and intrinsic determinants of the phase resetting curve for weak coupling

A phase resetting curve (PRC) keeps track of the extent to which a perturbation at a given phase advances or delays the next spike, and can be used to predict phase locking in networks of oscillators. The PRC can be estimated by convolving the waveform of the perturbation with the infinitesimal PRC (iPRC) under the assumption of weak coupling. The iPRC is often defined with respect to an infinitesimal current as z(i)(ϕ), where ϕ is phase, but can also be defined with respect to an infinitesimal conductance change as z(g)(ϕ). In this paper, we first show that the two approaches are equivalent. Coupling waveforms corresponding to synapses with different time courses sample z(g)(ϕ) in predictably different ways. We show that for oscillators with Type I excitability, an anomalous region in z(g)(ϕ) with opposite sign to that seen otherwise is often observed during an action potential. If the duration of the synaptic perturbation is such that it effectively samples this region, PRCs with both advances and delays can be observed despite Type I excitability. We also show that changing the duration of a perturbation so that it preferentially samples regions of stable or unstable slopes in z(g)(ϕ) can stabilize or destabilize synchrony in a network with the corresponding dynamics.

Inclusion of noise in iterated firing time maps based on the phase response curve

The infinitesimal phase response curve (PRC) of a neural oscillator to a weak input is a powerful predictor of network dynamics; however, many networks have strong coupling and require direct measurement of the PRC for strong inputs under the assumption of pulsatile coupling. We incorporate measured noise levels in firing time maps constructed from PRCs to predict phase-locked modes of activity, phase difference, and locking strength in 78 heterogeneous hybrid networks of 2 neurons constructed using the dynamic clamp. We show that noise may either destroy or stabilize a phase-locked mode of activity.

Pulse coupled oscillators and the phase resetting curve

Limit cycle oscillators that are coupled in a pulsatile manner are referred to as pulse coupled oscillators. In these oscillators, the interactions take the form of brief pulses such that the effect of one input dies out before the next is received. A phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike in an oscillatory neuron depending upon where in the cycle the input is applied. PRCs can be used to predict phase locking in networks of pulse coupled oscillators. In some studies of pulse coupled oscillators, a specific form is assumed for the interactions between oscillators, but a more general approach is to formulate the problem assuming a PRC that is generated using a perturbation that approximates the input received in the real biological network. In general, this approach requires that circuit architecture and a specific firing pattern be assumed. This allows the construction of discrete maps from one event to the next. The fixed points of these maps correspond to periodic firing modes and are easier to locate and analyze for stability compared to locating and analyzing periodic modes in the original network directly. Alternatively, maps based on the PRC have been constructed that do not presuppose a firing order. Specific circuits that have been analyzed under the assumption of pulsatile coupling include one to one lockings in a periodically forced oscillator or an oscillator forced at a fixed delay after a threshold event, two bidirectionally coupled oscillators with and without delays, a unidirectional N-ring of oscillators, and N all-to-all networks.

A new approach for determining phase response curves reveals that Purkinje cells can act as perfect integrators

Cerebellar Purkinje cells display complex intrinsic dynamics. They fire spontaneously, exhibit bistability, and via mutual network interactions are involved in the generation of high frequency oscillations and travelling waves of activity. To probe the dynamical properties of Purkinje cells we measured their phase response curves (PRCs). PRCs quantify the change in spike phase caused by a stimulus as a function of its temporal position within the interspike interval, and are widely used to predict neuronal responses to more complex stimulus patterns. Significant variability in the interspike interval during spontaneous firing can lead to PRCs with a low signal-to-noise ratio, requiring averaging over thousands of trials. We show using electrophysiological experiments and simulations that the PRC calculated in the traditional way by sampling the interspike interval with brief current pulses is biased. We introduce a corrected approach for calculating PRCs which eliminates this bias. Using our new approach, we show that Purkinje cell PRCs change qualitatively depending on the firing frequency of the cell. At high firing rates, Purkinje cells exhibit single-peaked, or monophasic PRCs. Surprisingly, at low firing rates, Purkinje cell PRCs are largely independent of phase, resembling PRCs of ideal non-leaky integrate-and-fire neurons. These results indicate that Purkinje cells can act as perfect integrators at low firing rates, and that the integration mode of Purkinje cells depends on their firing rate.

By observing how brief current pulses injected at different times between spikes change the phase of spiking of a neuron (and thus obtaining the so-called phase response curve), it should be possible to predict a full spike train in response to more complex stimulation patterns. When we applied this traditional protocol to obtain phase response curves in cerebellar Purkinje cells in the presence of noise, we observed a triangular region devoid of data points near the end of the spiking cycle. This “Bermuda Triangle” revealed a flaw in the classical method for constructing phase response curves. We developed a new approach to eliminate this flaw and used it to construct phase response curves of Purkinje cells over a range of spiking rates. Surprisingly, at low firing rates, phase changes were independent of the phase of the injected current pulses, implying that the Purkinje cell is a perfect integrator under these conditions. This mechanism has not yet been described in other cell types and may be crucial for the information processing capabilities of these neurons.

Phase response curve analysis of a full morphological globus pallidus neuron model reveals distinct perisomatic and dendritic modes of synaptic integration

Synchronization of globus pallidus (GP) neurons and cortically entrained oscillations between GP and other basal ganglia nuclei are key features of the pathophysiology of Parkinson's disease. Phase response curves (PRCs), which tabulate the effects of phasic inputs within a neuron's spike cycle on output spike timing, are efficient tools for predicting the emergence of synchronization in neuronal networks and entrainment to periodic input. In this study we apply physiologically realistic synaptic conductance inputs to a full morphological GP neuron model to determine the phase response properties of the soma and different regions of the dendritic tree. We find that perisomatic excitatory inputs delivered throughout the interspike interval advance the phase of the spontaneous spike cycle yielding a type I PRC. In contrast, we demonstrate that distal dendritic excitatory inputs can either delay or advance the next spike depending on whether they occur early or late in the spike cycle. We find this latter pattern of responses, summarized by a biphasic (type II) PRC, was a consequence of dendritic activation of the small conductance calcium-activated potassium current, SK. We also evaluate the spike-frequency dependence of somatic and dendritic PRC shapes, and we demonstrate the robustness of our results to variations of conductance densities, distributions, and kinetic parameters. We conclude that the distal dendrite of GP neurons embodies a distinct dynamical subsystem that could promote synchronization of pallidal networks to excitatory inputs. These results highlight the need to consider different effects of perisomatic and dendritic inputs in the control of network behavior.

Phase resetting curves allow for simple and accurate prediction of robust N:1 phase locking for strongly coupled neural oscillators

Existence and stability criteria for harmonic locking modes were derived for two reciprocally pulse coupled oscillators based on their first and second order phase resetting curves. Our theoretical methods are general in the sense that no assumptions about the strength of coupling, type of synaptic coupling, and model are made. These methods were then tested using two reciprocally inhibitory Wang and Buzsáki model neurons. The existence of bands of 2:1, 3:1, 4:1, and 5:1 phase locking in the relative frequency parameter space was predicted correctly, as was the phase of the slow neuron's spike within the cycle of the fast neuron in which it occurred. For weak coupling the bands are very narrow, but strong coupling broadens the bands. The predictions of the pulse coupled method agreed with weak coupling methods in the weak coupling regime, but extended predictability into the strong coupling regime. We show that our prediction method generalizes to pairs of neural oscillators coupled through excitatory synapses, and to networks of multiple oscillatory neurons. The main limitation of the method is the central assumption that the effect of each input dies out before the next input is received.

Functional phase response curves: a method for understanding synchronization of adapting neurons

Phase response curves (PRCs) for a single neuron are often used to predict the synchrony of mutually coupled neurons. Previous theoretical work on pulse-coupled oscillators used single-pulse perturbations. We propose an alternate method in which functional PRCs (fPRCs) are generated using a train of pulses applied at a fixed delay after each spike, with the PRC measured when the phasic relationship between the stimulus and the subsequent spike in the neuron has converged. The essential information is the dependence of the recovery time from pulse onset until the next spike as a function of the delay between the previous spike and the onset of the applied pulse. Experimental fPRCs in Aplysia pacemaker neurons were different from single-pulse PRCs, principally due to adaptation. In the biological neuron, convergence to the fully adapted recovery interval was slower at some phases than that at others because the change in the effective intrinsic period due to adaptation changes the effective phase resetting in a way that opposes and slows the effects of adaptation. The fPRCs for two isolated adapting model neurons were used to predict the existence and stability of 1:1 phase-locked network activity when the two neurons were coupled. A stability criterion was derived by linearizing a coupled map based on the fPRC and the existence and stability criteria were successfully tested in two-simulated-neuron networks with reciprocal inhibition or excitation. The fPRC is the first PRC-based tool that can account for adaptation in analyzing networks of neural oscillators.

Predictions of phase-locking in excitatory hybrid networks: excitation does not promote phase-locking in pattern-generating networks as reliably as inhibition

Phase-locked activity is thought to underlie many high-level functions of the nervous system, the simplest of which are produced by central pattern generators (CPGs). It is not known whether we can define a theoretical framework that is sufficiently general to predict phase-locking in actual biological CPGs, nor is it known why the CPGs that have been characterized are dominated by inhibition. Previously, we applied a method based on phase response curves measured using inputs of biologically realistic amplitude and duration to predict the existence and stability of 1:1 phase-locked modes in hybrid networks of one biological and one model bursting neuron reciprocally connected with artificial inhibitory synapses. Here we extend this analysis to excitatory coupling. Using the pyloric dilator neuron from the stomatogastric ganglion of the American lobster as our biological cell, we experimentally prepared 86 networks using five biological neurons, four model neurons, and heterogeneous synapse strengths between 1 and 10,000 nS. In 77% of networks, our method was robust to biological noise and accurately predicted the phasic relationships. In 3%, our method was inaccurate. The remaining 20% were not amenable to analysis because our theoretical assumptions were violated. The high failure rate for excitation compared with inhibition was due to differential effects of noise and feedback on excitatory versus inhibitory coupling and suggests that CPGs dominated by excitatory synapses would require precise tuning to function, which may explain why CPGs rely primarily on inhibitory synapses.