Using phase resetting to predict 1:1 and 2:2 locking in two neuron networks in which firing order is not always preserved

Our goal is to understand how nearly synchronous modes arise in heterogenous networks of neurons. In heterogenous networks, instead of exact synchrony, nearly synchronous modes arise, which include both 1:1 and 2:2 phase-locked modes. Existence and stability criteria for 2:2 phase-locked modes in reciprocally coupled two neuron circuits were derived based on the open loop phase resetting curve (PRC) without the assumption of weak coupling. The PRC for each component neuron was generated using the change in synaptic conductance produced by a presynaptic action potential as the perturbation. Separate derivations were required for modes in which the firing order is preserved and for those in which it alternates. Networks composed of two model neurons coupled by reciprocal inhibition were examined to test the predictions. The parameter regimes in which both types of nearly synchronous modes are exhibited were accurately predicted both qualitatively and quantitatively provided that the synaptic time constant is short with respect to the period and that the effect of second order resetting is considered. In contrast, PRC methods based on weak coupling could not predict 2:2 modes and did not predict the 1:1 modes with the level of accuracy achieved by the strong coupling methods. The strong coupling prediction methods provide insight into what manipulations promote near-synchrony in a two neuron network and may also have predictive value for larger networks, which can also manifest changes in firing order. We also identify a novel route by which synchrony is lost in mildly heterogenous networks.     

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.     

Enhancement of spike-timing precision by autaptic transmission in neocortical inhibitory interneurons

In vivo studies suggest that precise firing of neurons is important for correct sensory representation. Principal neocortical neurons fire imprecisely when repeatedly activated by fixed sensory stimuli or current depolarizations. Here we show that in contrast to pyramidal neurons, firing in neocortical GABAergic fast-spiking (FS) interneurons is quite precise. FS interneurons are self-innervated by powerful GABAergic autaptic connections reliably activated after each spike, suggesting that autapses strongly regulate FS-cell spike timing. Indeed, blockade of autaptic transmission degraded temporal precision in multiple ways. Under these conditions, realistic dynamic-clamp hyperpolarizing autapses restored precision of spike timing, even in the presence of synaptic noise. Furthermore, firing precision was increased in pyramidal neurons by artificial GABAergic autaptic conductances, suggesting that tightly coupled synaptic feedback inhibition regulates spike timing in principal cells. Thus, well-timed inhibition, whether autaptic or synaptic, facilitates precise spike timing and promotes synchronized cortical network oscillations relevant to several behaviors.     

Influences of membrane properties on phase response curve and synchronization stability in a model globus pallidus neuron

The activity patterns of the globus pallidus (GPe) and subthalamic nucleus (STN) are closely associated with motor function and dysfunction in the basal ganglia. In the pathological state caused by dopamine depletion, the STN–GPe network exhibits rhythmic synchronous activity accompanied by rebound bursts in the STN. Therefore, the mechanism of activity transition is a key to understand basal ganglia functions. As synchronization in GPe neurons could induce pathological STN rebound bursts, it is important to study how synchrony is generated in the GPe. To clarify this issue, we applied the phase-reduction technique to a conductance-based GPe neuronal model in order to derive the phase response curve (PRC) and interaction function between coupled GPe neurons. Using the PRC and interaction function, we studied how the steady-state activity of the GPe network depends on intrinsic membrane properties, varying ionic conductances on the membrane. We noted that a change in persistent sodium current, fast delayed rectifier Kv3 potassium current, M-type potassium current and small conductance calcium-dependent potassium current influenced the PRC shape and the steady state. The effect of those currents on the PRC shape could be attributed to extension of the firing period and reduction of the phase response immediately after an action potential. In particular, the slow potassium current arising from the M-type potassium and the SK current was responsible for the reduction of the phase response. These results suggest that the membrane property modulation controls synchronization/asynchronization in the GPe and the pathological pattern of STN–GPe activity.     

Spike generating dynamics and the conditions for spike-time precision in cortical neurons

Temporal precision of spiking response in cortical neurons has been a subject of intense debate. Using a canonical model of spike generation, we explore the conditions for precise and reliable spike timing in the presence of Gaussian white noise. In agreement with previous results we find that constant stimuli lead to imprecise timing, while aperiodic stimuli yield precise spike timing. Under constant stimulus the neuron is a noise perturbed oscillator, the spike times follow renewal statistics and are imprecise. Under an aperiodic stimulus sequence, the neuron acts as a threshold element; the firing times are precisely determined by the dynamics of the stimulus. We further study the dependence of spike-time precision on the input stimulus frequency and find a non-linear tuning whose width can be related to the locking modes of the neuron. We conclude that viewing the neuron as a non-linear oscillator is the key for understanding spike-time precision.     

Phantoms in the brain: Ambiguous representations of stimulus amplitude and timing in weakly electric fish

In wave-type weakly electric fish, two distinct types of primary afferent fibers are specialized for separately encoding modulations in the amplitude and phase (timing) of electrosensory stimuli. Time-coding afferents phase lock to periodic stimuli and respond to changes in stimulus phase with shifts in spike timing. Amplitude-coding afferents fire sporadically to periodic stimuli. Their probability of firing in a given cycle, and therefore their firing rate, is proportional to stimulus amplitude. However, the spike times of time-coding afferents are also affected by changes in amplitude; similarly, the firing rates of amplitude-coding afferents are also affected by changes in phase. Because identical changes in the activity of an individual primary afferent can be caused by modulations in either the amplitude or phase of stimuli, there is ambiguity regarding the information content of primary afferent responses that can result in ‘phantom’ modulations not present in an actual stimulus. Central electrosensory neurons in the hindbrain and midbrain respond to these phantom modulations. Phantom modulations can also elicit behavioral responses, indicating that ambiguity in the encoding of amplitude and timing information ultimately distorts electrosensory perception. A lack of independence in the encoding of multiple stimulus attributes can therefore result in perceptual illusions. Similar effects may occur in other sensory systems as well. In particular, the vertebrate auditory system is thought to be phylogenetically related to the electrosensory system and it encodes information about amplitude and timing in similar ways. It has been well established that pitch perception and loudness perception are both affected by the frequency and intensity of sounds, raising the intriguing possibility that auditory perception may also be affected by ambiguity in the encoding of sound amplitude and timing.     

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.     

Variability and coding efficiency of noisy neural spike encoders

Encoding synaptic inputs as a train of action potentials is a fundamental function of nerve cells. Although spike trains recorded in vivo have been shown to be highly variable, it is unclear whether variability in spike timing represents faithful encoding of temporally varying synaptic inputs or noise inherent in the spike encoding mechanism. It has been reported that spike timing variability is more pronounced for constant, unvarying inputs than for inputs with rich temporal structure. This could have significant implications for the nature of neural coding, particularly if precise timing of spikes and temporal synchrony between neurons is used to represent information in the nervous system. To study the potential functional role of spike timing variability, we estimate the fraction of spike timing variability which conveys information about the input for two types of noisy spike encoders--an integrate and fire model with randomly chosen thresholds and a model of a patch of neuronal membrane containing stochastic Na(+) and K(+) channels obeying Hodgkin-Huxley kinetics. The quality of signal encoding is assessed by reconstructing the input stimuli from the output spike trains using optimal linear mean square estimation. A comparison of the estimation performance of noisy neuronal models of spike generation enables us to assess the impact of neuronal noise on the efficacy of neural coding. The results for both models suggest that spike timing variability reduces the ability of spike trains to encode rapid time-varying stimuli. Moreover, contrary to expectations based on earlier studies, we find that the noisy spike encoding models encode slowly varying stimuli more effectively than rapidly varying ones.     

The chronotron: a neuron that learns to fire temporally precise spike patterns

In many cases, neurons process information carried by the precise timings of spikes. Here we show how neurons can learn to generate specific temporally precise output spikes in response to input patterns of spikes having precise timings, thus processing and memorizing information that is entirely temporally coded, both as input and as output. We introduce two new supervised learning rules for spiking neurons with temporal coding of information (chronotrons), one that provides high memory capacity (E-learning), and one that has a higher biological plausibility (I-learning). With I-learning, the neuron learns to fire the target spike trains through synaptic changes that are proportional to the synaptic currents at the timings of real and target output spikes. We study these learning rules in computer simulations where we train integrate-and-fire neurons. Both learning rules allow neurons to fire at the desired timings, with sub-millisecond precision. We show how chronotrons can learn to classify their inputs, by firing identical, temporally precise spike trains for different inputs belonging to the same class. When the input is noisy, the classification also leads to noise reduction. We compute lower bounds for the memory capacity of chronotrons and explore the influence of various parameters on chronotrons' performance. The chronotrons can model neurons that encode information in the time of the first spike relative to the onset of salient stimuli or neurons in oscillatory networks that encode information in the phases of spikes relative to the background oscillation. Our results show that firing one spike per cycle optimizes memory capacity in neurons encoding information in the phase of firing relative to a background rhythm.     

Phase resetting and phase locking in hybrid circuits of one model and one biological neuron

To determine why elements of central pattern generators phase lock in a particular pattern under some conditions but not others, we tested a theoretical pattern prediction method. The method is based on the tabulated open loop pulsatile interactions of bursting neurons on a cycle-by-cycle basis and was tested in closed loop hybrid circuits composed of one bursting biological neuron and one bursting model neuron coupled using the dynamic clamp. A total of 164 hybrid networks were formed by varying the synaptic conductances. The prediction of 1:1 phase locking agreed qualitatively with the experimental observations, except in three hybrid circuits in which 1:1 locking was predicted but not observed. Correct predictions sometimes required consideration of the second order phase resetting, which measures the change in the timing of the second burst after the perturbation. The method was robust to offsets between the initiation of bursting in the presynaptic neuron and the activation of the synaptic coupling with the postsynaptic neuron. The quantitative accuracy of the predictions fell within the variability (10%) in the experimentally observed intrinsic period and phase resetting curve (PRC), despite changes in the burst duration of the neurons between open and closed loop conditions.     

Control of multistability in ring circuits of oscillators

The essential dynamics of some biological central pattern generators (CPGs) can be captured by a model consisting of N neurons connected in a ring. These circuits, like many oscillatory nonlinear circuits of sufficient complexity, are capable of multistability, that is, of generating different firing patterns distinguished by the phasic relationships between the firing in each circuit element (neuron). Moreover, a shift in firing pattern can be induced by a transient perturbation. A systematic approach, based on phase-response curve (PRC) theory, was used to determine the optimum timing for perturbations that induce a shift in the firing pattern. The first step was to visualize the solution space of the ring circuit, including the attractive basins for each stable firing pattern; this was possible using the relative phase of N−1 oscillators, with respect to an arbitrarily selected reference oscillator, as coordinate axes. The trajectories in this phase space were determined using an iterative mapping based only on the PRCs of the uncoupled component oscillators; this algorithm was called a circuit emulator. For an accurate mapping of the attractive basin of each pattern exhibited by the ring circuit, the emulator had to take into account the effect of a perturbation or input on the timing of two bursts following the onset of the perturbation, rather than just one. The visualization of the attractive basins for rings of two, three, and four oscillators enabled the accurate prediction of the amounts of phase resetting applied to up to N−1 oscillators within a cycle that would induce a transition from any pattern to any another pattern. Finally, the timing and synaptic characterization of an input called the switch signal was adjusted to produce the desired amount of phase resetting. Received: 29 May 1998 / Accepted in revised form: 18 September 1998     

Impact of adaptation currents on synchronization of coupled exponential integrate-and-fire neurons

The ability of spiking neurons to synchronize their activity in a network depends on the response behavior of these neurons as quantified by the phase response curve (PRC) and on coupling properties. The PRC characterizes the effects of transient inputs on spike timing and can be measured experimentally. Here we use the adaptive exponential integrate-and-fire (aEIF) neuron model to determine how subthreshold and spike-triggered slow adaptation currents shape the PRC. Based on that, we predict how synchrony and phase locked states of coupled neurons change in presence of synaptic delays and unequal coupling strengths. We find that increased subthreshold adaptation currents cause a transition of the PRC from only phase advances to phase advances and delays in response to excitatory perturbations. Increased spike-triggered adaptation currents on the other hand predominantly skew the PRC to the right. Both adaptation induced changes of the PRC are modulated by spike frequency, being more prominent at lower frequencies. Applying phase reduction theory, we show that subthreshold adaptation stabilizes synchrony for pairs of coupled excitatory neurons, while spike-triggered adaptation causes locking with a small phase difference, as long as synaptic heterogeneities are negligible. For inhibitory pairs synchrony is stable and robust against conduction delays, and adaptation can mediate bistability of in-phase and anti-phase locking. We further demonstrate that stable synchrony and bistable in/anti-phase locking of pairs carry over to synchronization and clustering of larger networks. The effects of adaptation in aEIF neurons on PRCs and network dynamics qualitatively reflect those of biophysical adaptation currents in detailed Hodgkin-Huxley-based neurons, which underscores the utility of the aEIF model for investigating the dynamical behavior of networks. Our results suggest neuronal spike frequency adaptation as a mechanism synchronizing low frequency oscillations in local excitatory networks, but indicate that inhibition rather than excitation generates coherent rhythms at higher frequencies.     

On the Firing Rate Dependency of the Phase Response Curve of Rat Purkinje Neurons In Vitro

Synchronous spiking during cerebellar tasks has been observed across Purkinje cells: however, little is known about the intrinsic cellular mechanisms responsible for its initiation, cessation and stability. The Phase Response Curve (PRC), a simple input-output characterization of single cells, can provide insights into individual and collective properties of neurons and networks, by quantifying the impact of an infinitesimal depolarizing current pulse on the time of occurrence of subsequent action potentials, while a neuron is firing tonically. Recently, the PRC theory applied to cerebellar Purkinje cells revealed that these behave as phase-independent integrators at low firing rates, and switch to a phase-dependent mode at high rates. Given the implications for computation and information processing in the cerebellum and the possible role of synchrony in the communication with its post-synaptic targets, we further explored the firing rate dependency of the PRC in Purkinje cells. We isolated key factors for the experimental estimation of the PRC and developed a closed-loop approach to reliably compute the PRC across diverse firing rates in the same cell. Our results show unambiguously that the PRC of individual Purkinje cells is firing rate dependent and that it smoothly transitions from phase independent integrator to a phase dependent mode. Using computational models we show that neither channel noise nor a realistic cell morphology are responsible for the rate dependent shift in the phase response curve.     

Existence and stability criteria for phase-locked modes in ring neural networks based on the spike time resetting curve method

We developed a systematic and consistent mathematical approach to predicting 1:1 phase-locked modes in ring neural networks of spiking neurons based on the open loop spike time resetting curve (STRC) and its almost equivalent counterpart—the phase resetting curve (PRC). The open loop STRCs/PRCs were obtained by injecting into an isolated model neuron a triangular shaped time-dependent stimulus current closely resembling an actual synaptic input. Among other advantages, the STRC eliminates the confusion regarding the undefined phase for stimuli driving the neuron outside of the unperturbed limit cycle. We derived both open loop PRC and STRC-based existence and stability criteria for 1:1 phase-locked modes developed in ring networks of spiking neurons. Our predictions were in good agreement with the closed loop numerical simulations. Intuitive graphical methods for predicting phase-locked modes were also developed both for half-centers and for larger ring networks.     

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.     

Complementary responses to mean and variance modulations in the perfect integrate-and-fire model

In the perfect integrate-and-fire model (PIF), the membrane voltage is proportional to the integral of the input current since the time of the previous spike. It has been shown that the firing rate within a noise free ensemble of PIF neurons responds instantaneously to dynamic changes in the input current, whereas in the presence of white noise, model neurons preferentially pass low frequency modulations of the mean current. Here, we prove that when the input variance is perturbed while holding the mean current constant, the PIF responds preferentially to high frequency modulations. Moreover, the linear filters for mean and variance modulations are complementary, adding exactly to one. Since changes in the rate of Poisson distributed inputs lead to proportional changes in the mean and variance, these results imply that an ensemble of PIF neurons transmits a perfect replica of the time-varying input rate for Poisson distributed input. A more general argument shows that this property holds for any signal leading to proportional changes in the mean and variance of the input current.     

Cholinergic Neuromodulation Changes Phase Response Curve Shape and Type in Cortical Pyramidal Neurons

Spike generation in cortical neurons depends on the interplay between diverse intrinsic conductances. The phase response curve (PRC) is a measure of the spike time shift caused by perturbations of the membrane potential as a function of the phase of the spike cycle of a neuron. Near the rheobase, purely positive (type I) phase-response curves are associated with an onset of repetitive firing through a saddle-node bifurcation, whereas biphasic (type II) phase-response curves point towards a transition based on a Hopf-Andronov bifurcation. In recordings from layer 2/3 pyramidal neurons in cortical slices, cholinergic action, consistent with down-regulation of slow voltage-dependent potassium currents such as the M-current, switched the PRC from type II to type I. This is the first report showing that cholinergic neuromodulation may cause a qualitative switch in the PRCs type implying a change in the fundamental dynamical mechanism of spike generation.     

Millisecond-Scale Motor Encoding in a Cortical Vocal Area

Studies of motor control have almost universally examined firing rates to investigate how the brain shapes behavior. In principle, however, neurons could encode information through the precise temporal patterning of their spike trains as well as (or instead of) through their firing rates. Although the importance of spike timing has been demonstrated in sensory systems, it is largely unknown whether timing differences in motor areas could affect behavior. We tested the hypothesis that significant information about trial-by-trial variations in behavior is represented by spike timing in the songbird vocal motor system. We found that neurons in motor cortex convey information via spike timing far more often than via spike rate and that the amount of information conveyed at the millisecond timescale greatly exceeds the information available from spike counts. These results demonstrate that information can be represented by spike timing in motor circuits and suggest that timing variations evoke differences in behavior.     

Dynamical features of higher-order correlation events: impact on cortical cells

Cortical neurons receive signals from thousands of other neurons. The statistical properties of the input spike trains substantially shape the output response properties of each neuron. Experimental and theoretical investigations have mostly focused on the second order statistical features of the input spike trains (mean firing rates and pairwise correlations). Little is known of how higher order correlations affect the integration and firing behavior of a cell independently of the second order statistics. To address this issue, we simulated the dynamics of a population of 5000 neurons, controlling both their second order and higher-order correlation properties to reflect physiological data. We then used these ensemble dynamics as the input stage to morphologically reconstructed cortical cells (layer 5 pyramidal, layer 4 spiny stellate cell), and to an integrate and fire neuron. Our results show that changes done solely to the higher-order correlation properties of the network’s dynamics significantly affect the response properties of a target neuron, both in terms of output rate and spike timing. Moreover, the neuronal morphology and voltage dependent mechanisms of the target neuron considerably modulate the quantitative aspects of these effects. Finally, we show how these results affect sparseness of neuronal representations, tuning properties, and feature selectivity of cortical cells. An erratum to this article can be found at     

Spike latency and jitter of neuronal membrane patches with stochastic Hodgkin-Huxley channels

Ion channel stochasticity can influence the voltage dynamics of neuronal membrane, with stronger effects for smaller patches of membrane because of the correspondingly smaller number of channels. We examine this question with respect to first spike statistics in response to a periodic input of membrane patches including stochastic Hodgkin-Huxley channels, comparing these responses to spontaneous firing. Without noise, firing threshold of the model depends on frequency—a sinusoidal stimulus is subthreshold for low and high frequencies and suprathreshold for intermediate frequencies. When channel noise is added, a stimulus in the lower range of subthreshold frequencies can influence spike output, while high subthreshold frequencies remain subthreshold. Both input frequency and channel noise strength influence spike timing. Specifically, spike latency and jitter have distinct minima as a function of input frequency, showing a resonance like behavior. With either no input, or low frequency subthreshold input, or input in the low or high suprathreshold frequency range, channel noise reduces latency and jitter, with the strongest impact for the lowest input frequencies. In contrast, for an intermediate range of suprathreshold frequencies, where an optimal input gives a minimum latency, the noise effect reverses, and spike latency and jitter increase with channel noise. Thus, a resonant minimum of the spike response as a function of frequency becomes more pronounced with less noise. Spike latency and jitter also depend on the initial phase of the input, resulting in minimal latencies at an optimal phase, and depend on the membrane time constant, with a longer time constant broadening frequency tuning for minimal latency and jitter. Taken together, these results suggest how stochasticity of ion channels may influence spike timing and thus coding for neurons with functionally localized concentrations of channels, such as in “hot spots” of dendrites, spines or axons.