Precision and reliability of periodically and quasiperiodically driven integrate-and-fire neurons

Beyond the Bristol book: Advances and perspectives in non-smooth dynamics and applications

Non-smooth dynamics induced by switches, impacts, sliding, and other abrupt changes are pervasive in physics, biology, and engineering. Yet, systems with non-smooth dynamics have historically received far less attention compared to their smooth counterparts. The classic "Bristol book" [di Bernardo et al., Piecewise-smooth Dynamical Systems. Theory and Applications (Springer-Verlag, 2008)] contains a 2008 state-of-the-art review of major results and challenges in the study of non-smooth dynamical systems. In this paper, we provide a detailed review of progress made since 2008. We cover hidden dynamics, generalizations of sliding motion, the effects of noise and randomness, multi-scale approaches, systems with time-dependent switching, and a variety of local and global bifurcations. Also, we survey new areas of application, including neuroscience, biology, ecology, climate sciences, and engineering, to which the theory has been applied.

Resonance in the Mouse Ventral Tegmental Area Dopaminergic Network Induced by Regular and Poisson Distributed Optogenetic Stimulation in-vitro

Neurons in many brain regions exhibit spontaneous, intrinsic rhythmic firing activity. This rhythmic firing activity may determine the way in which these neurons respond to extrinsic synaptic inputs. We hypothesized that neurons should be most responsive to inputs at the frequency of the intrinsic oscillation frequency. We addressed this question in the ventral tegmental area (VTA), a dopaminergic nucleus in the midbrain. VTA neurons have a unique propensity to exhibit spontaneous intrinsic rhythmic activity in the 1-5 Hz frequency range, which persists in the in-vitro brain slice, and form a network of weakly coupled oscillators. Here, we combine in-vitro simultaneous recording of action potentials from a 60 channel multi-electrode-array with cell-type-specific optogenetic stimulation of the VTA dopamine neurons. We investigated how VTA neurons respond to wide-band stochastic (Poisson input) as well as regular laser pulses. Strong synchrony was induced between the laser input and the spike timing of the neurons, both for regular pulse trains and Poisson pulse trains. For rhythmically pulsed input, the neurons demonstrated resonant behavior with the strongest phase locking at their intrinsic oscillation frequency, but also at half and double the intrinsic oscillation frequency. Stochastic Poisson pulse stimulation provided a more effective stimulation of the entire population, yet we observed resonance at lower frequencies (approximately half the oscillation frequency) than the neurons' intrinsic oscillation frequency. The non-linear filter characteristics of dopamine neurons could allow the VTA to predict precisely timed future rewards based on past sensory inputs, a crucial component of reward prediction error signaling. In addition, these filter characteristics could contribute to a pacemaker role for the VTA in synchronizing activity with other regions like the prefrontal cortex and the hippocampus.

Rhythmic modulation of thalamic oscillations depends on intrinsic cellular dynamics

OBJECTIVE

Rhythmic brain stimulation has emerged as a powerful tool to modulate cognition and to target pathological oscillations related to neurological and psychiatric disorders. However, we lack a systematic understanding of how periodic stimulation interacts with endogenous neural activity as a function of the brain state and target.

APPROACH

To address this critical issue, we applied periodic stimulation to a unified biophysical thalamic network model that generates multiple distinct oscillations, and examined thoroughly the impact of rhythmic stimulation on different oscillatory states.

MAIN RESULTS

We found that rhythmic perturbation induces four basic response mechanisms: entrainment, acceleration, resonance and suppression. Importantly, the appearance and expression of these mechanisms depend highly on the intrinsic cellular dynamics in each state. Specifically, the low-threshold bursting of thalamocortical cells (TCs) in delta (δ) oscillation renders the network relatively insensitive to entrainment; the high-threshold bursting of TCs in alpha (α) oscillation leads to widespread oscillation suppression while the tonic spiking of TC cells in gamma (γ) oscillation results in prominent entrainment and resonance. In addition, we observed entrainment discontinuity during α oscillation that is mediated by firing pattern switching of high-threshold bursting TC cells. Furthermore, we demonstrate that direct excitatory stimulation of the lateral geniculate nucleus (LGN) entrains thalamic oscillations via an asymmetric Arnold tongue that favors higher frequency entrainment and resonance, while stimulation of the inhibitory circuit, the reticular nucleus, induces much weaker and more symmetric entrainment and resonance. These results support the notion that rhythmic stimulation engages brain oscillations in a state- and target-dependent manner.

SIGNIFICANCE

Overall, our study provides, for the first time, insights into how the biophysics of thalamic oscillations guide the emergence of complex, state-dependent mechanisms of target engagement, which can be leveraged for the future rational design of novel therapeutic stimulation modalities.

Unified thalamic model generates multiple distinct oscillations with state-dependent entrainment by stimulation

The thalamus plays a critical role in the genesis of thalamocortical oscillations, yet the underlying mechanisms remain elusive. To understand whether the isolated thalamus can generate multiple distinct oscillations, we developed a biophysical thalamic model to test the hypothesis that generation of and transition between distinct thalamic oscillations can be explained as a function of neuromodulation by acetylcholine (ACh) and norepinephrine (NE) and afferent synaptic excitation. Indeed, the model exhibited four distinct thalamic rhythms (delta, sleep spindle, alpha and gamma oscillations) that span the physiological states corresponding to different arousal levels from deep sleep to focused attention. Our simulation results indicate that generation of these distinct thalamic oscillations is a result of both intrinsic oscillatory cellular properties and specific network connectivity patterns. We then systematically varied the ACh/NE and input levels to generate a complete map of the different oscillatory states and their transitions. Lastly, we applied periodic stimulation to the thalamic network and found that entrainment of thalamic oscillations is highly state-dependent. Our results support the hypothesis that ACh/NE modulation and afferent excitation define thalamic oscillatory states and their response to brain stimulation. Our model proposes a broader and more central role of the thalamus in the genesis of multiple distinct thalamo-cortical rhythms than previously assumed.

Computational modeling has served as an important tool to understand the cellular and circuit mechanisms of thalamocortical oscillations. However, most of the existing thalamic models focus on only one particular oscillatory pattern such as alpha or spindle oscillations. Thus, it remains unclear whether the same thalamic circuitry on its own could generate all major oscillatory patterns and if so what mechanisms underlie the transition among these distinct states. Here we present a unified model of the thalamus that is capable of independently generating multiple distinct oscillations corresponding to different physiological conditions. We then mapped out the different thalamic oscillations by varying the ACh/NE modulatory level and input level systematically. Our simulation results offer a mechanistic understanding of thalamic oscillations and support the long standing notion of a thalamic “pacemaker”. It also suggests that pathological oscillations associated with neurological and psychiatric disorders may stem from malfunction of the thalamic circuitry.

Phase Difference between Model Cortical Areas Determines Level of Information Transfer

Communication between cortical sites is mediated by long-range synaptic connections. However, these connections are relatively static, while everyday cognitive tasks demand a fast and flexible routing of information in the brain. Synchronization of activity between distant cortical sites has been proposed as the mechanism underlying such a dynamic communication structure. Here, we study how oscillatory activity affects the excitability and input-output relation of local cortical circuits and how it alters the transmission of information between cortical circuits. To this end, we develop model circuits showing fast oscillations by the PING mechanism, of which the oscillatory characteristics can be altered. We identify conditions for synchronization between two brain circuits and show that the level of intercircuit coherence and the phase difference is set by the frequency difference between the intrinsic oscillations. We show that the susceptibility of the circuits to inputs, i.e., the degree of change in circuit output following input pulses, is not uniform throughout the oscillation period and that both firing rate, frequency and power are differentially modulated by inputs arriving at different phases. As a result, an appropriate phase difference between the circuits is critical for the susceptibility windows of the circuits in the network to align and for information to be efficiently transferred. We demonstrate that changes in synchrony and phase difference can be used to set up or abolish information transfer in a network of cortical circuits.

Spike-count distribution in a neuronal population under weak common stimulation

We study the probability distribution of the number of synchronous action potentials (spike count) in a model network consisting of a homogeneous neural population that is driven by a common time-dependent stimulus. We derive two analytical approximations for the count statistics, which are based on linear response theory and hold true for weak input correlations. Comparison to numerical simulations of populations of integrate-and-fire neurons in different parameter regimes reveals that our theory correctly predicts how much a weak common stimulus increases the probability of common firing and of common silence in the neural population.

Dynamical analysis of periodic bursting in piece-wise linear planar neuron model

A piece-wise linear planar neuron model, namely, two-dimensional McKean model with periodic drive is investigated in this paper. Periodical bursting phenomenon can be observed in the numerical simulations. By assuming the formal solutions associated with different intervals of this non-autonomous system and introducing the generalized Jacobian matrix at the non-smooth boundaries, the bifurcation mechanism for the bursting solution induced by the slowly varying periodic drive is presented. It is shown that, the discontinuous Hopf bifurcation occurring at the non-smooth boundaries, i.e., the bifurcation taking place at the thresholds of the stimulation, leads the alternation between the rest state and spiking state. That is, different oscillation modes of this non-autonomous system convert periodically due to the non-smoothness of the vector field and the slow variation of the periodic drive as well.

Dynamics and Reliability of Bistable Neurons Driven with Time-Dependent Stimuli

The reliability of a spiking neuron depends on the frequency content of the driving input signal. Previous studies have shown that well above threshold, regularly firing neurons generate reliable responses when the input signal resonates with the firing frequency of the cell. Instead, well below threshold, reliable responses are obtained when the input frequency resonates with the subthreshold oscillations of the neuron. Previous theories, however, provide no clear prediction for the input frequency giving rise to maximally reliable spiking at threshold, which is probably the most relevant firing regime in mammalian cortex under physiological conditions. In particular, when the firing onset is governed by a subcritical Hopf bifurcation, the frequency of subthreshold oscillations often differs from the firing rate at threshold. The predictions of previous studies, hence, cannot be smoothly merged at threshold. Here we explore the behavior of reliability in bistable neurons near threshold using three types of driving stimuli: constant, periodic, and stochastic. We find that the two natural frequencies of the system, associated with the two coexisting attractors, provide a rich variety of possible locking modes with the external signal. Reliability is determined by the sensitivity to noise of each locking mode and by the transition probabilities between modes. Noise increases the amount of spike time jitter, and minimal jitter is obtained for input frequencies coinciding with the suprathreshold firing rate of the cell. In addition, noise may either enhance or inhibit transitions between the two attractors, depending on the input frequency. The dual role played by noise in bistable systems implies that reliability is determined by a delicate balance between spike time jitter and the rate of transitions between attractors.

A transition to sharp timing in stochastic leaky integrate-and-fire neurons driven by frozen noisy input

The firing activity of intracellularly stimulated neurons in cortical slices has been demonstrated to be profoundly affected by the temporal structure of the injected current (Mainen & Sejnowski, 1995 ). This suggests that the timing features of the neural response may be controlled as much by its own biophysical characteristics as by how a neuron is wired within a circuit. Modeling studies have shown that the interplay between internal noise and the fluctuations of the driving input controls the reliability and the precision of neuronal spiking (Cecchi et al., 2000 ; Tiesinga, 2002 ; Fellous, Rudolph, Destexhe, & Sejnowski, 2003 ). In order to investigate this interplay, we focus on the stochastic leaky integrate-and-fire neuron and identify the Hölder exponent H of the integrated input as the key mathematical property dictating the regime of firing of a single-unit neuron. We have recently provided numerical evidence (Taillefumier & Magnasco, 2013 ) for the existence of a phase transition when [Formula: see text] becomes less than the statistical Hölder exponent associated with internal gaussian white noise (H=1/2). Here we describe the theoretical and numerical framework devised for the study of a neuron that is periodically driven by frozen noisy inputs with exponent H>0. In doing so, we account for the existence of a transition between two regimes of firing when H=1/2, and we show that spiking times have a continuous density when the Hölder exponent satisfies H>1/2. The transition at H=1/2 formally separates rate codes, for which the neural firing probability varies smoothly, from temporal codes, for which the neuron fires at sharply defined times regardless of the intensity of internal noise.

Rhythm-induced spike-timing patterns characterized by 1D firing maps

We explore patterns in the spike timing of neurons receiving periodic inputs, with an emphasis on stable characteristics which are realized in both models and in-vitro whole-cell recordings. We report on whole-cell recordings of pyramidal CA1 cells from rat hippocampus and entorhinal cortex and compare this data to model simulations. Cells were injected with a constant current to induce a steady firing rate and then a modest rhythm was added which altered the spike times and their corresponding phases relative to the rhythm. For both experiment and theory the relationship between consecutive spike phases is characterized by a probability distribution with peaks concentrated near a one-dimensional firing map. As is well-known, stable fixed points of this map correspond to the neuron phase-locking to the rhythm. We show that the interaction between noise and sufficiently steep maps can also cause a new kind of spike-time organization, in which consecutive spike time pairs organize into discrete clusters, with transitions between these clusters proceeding in a fixed sequence. This structure is not just a vestige of the noise-free dynamics. This slow dynamics and temporal organization in the relationship between consecutive spike phases is not evident in either the neuron's voltage traces or single phase or interspike interval histograms. Furthermore, the consecutive spike relationship is also evident in consecutive ISIs, and hence this ordering can be observed without detailed knowledge of the rhythm (e.g. without concurrent LFP recordings).

Coherence depression in stochastic excitable systems with two-frequency forcing

We study the response of two generic neuron models, the leaky integrate-and-fire (LIF) model and the leaky integrate-and-fire model with dynamic threshold (LIFDT) (i.e., with memory) to a stimulus consisting of two sinusoidal drives with incommensurate frequency, an amplitude modulation ("envelope") noise and a relatively weak additive noise. Spectral and coherence analysis of responses to such naturalistic stimuli reveals how the LIFDT model exhibits better correlation between modulation and spike train even in the presence of both noises. However, a resonance-induced synchrony, occurring when the beat frequency between the sinusoids is close to the intrinsic neuronal firing rate, decreases the coherence in the dynamic threshold case. Under suprathreshold conditions, the modulation noise simultaneously decreases the linear spectral coherence between the spikes and the whole stimulus, as well as between spikes and the stimulus envelope. Our study shows that the coefficient of variation of the envelope fluctuations is positively correlated with the degree of coherence depression. As the coherence function quantifies the linear information transmission, our findings indicate that under certain conditions, a transmission loss results when an excitable system with adaptive properties encodes a beat with frequency in the vicinity of its mean firing rate.

Neuronal spike-train responses in the presence of threshold noise

The variability of neuronal firing has been an intense topic of study for many years. From a modelling perspective it has often been studied in conductance based spiking models with the use of additive or multiplicative noise terms to represent channel fluctuations or the stochastic nature of neurotransmitter release. Here we propose an alternative approach using a simple leaky integrate-and-fire model with a noisy threshold. Initially, we develop a mathematical treatment of the neuronal response to periodic forcing using tools from linear response theory and use this to highlight how a noisy threshold can enhance downstream signal reconstruction. We further develop a more general framework for understanding the responses to large amplitude forcing based on a calculation of first passage times. This is ideally suited to understanding stochastic mode-locking, for which we numerically determine the Arnol'd tongue structure. An examination of data from regularly firing stellate neurons within the ventral cochlear nucleus, responding to sinusoidally amplitude modulated pure tones, shows tongue structures consistent with these predictions and highlights that stochastic, as opposed to deterministic, mode-locking is utilised at the level of the single stellate cell to faithfully encode periodic stimuli.

Mechanisms for Phase Shifting in Cortical Networks and their Role in Communication through Coherence

In the primate visual cortex, the phase of spikes relative to oscillations in the local field potential (LFP) in the gamma frequency range (30-80 Hz) can be shifted by stimulus features such as orientation and thus the phase may carry information about stimulus identity. According to the principle of communication through coherence (CTC), the relative LFP phase between the LFPs in the sending and receiving circuits affects the effectiveness of the transmission. CTC predicts that phase shifting can be used for stimulus selection. We review and investigate phase shifting in models of periodically driven single neurons and compare it with phase shifting in models of cortical networks. In a single neuron, as the driving current is increased, the spike phase varies systematically while the firing rate remains constant. In a network model of reciprocally connected excitatory (E) and inhibitory (I) cells phase shifting occurs in response to both injection of constant depolarizing currents and to brief pulses to I cells. These simple models provide an account for phase-shifting observed experimentally and suggest a mechanism for implementing CTC. We discuss how this hypothesis can be tested experimentally using optogenetic techniques.

Spatial attention in area V4 is mediated by circuits in primary visual cortex

The ability to covertly select visual stimuli in our environment based on their behavioral relevance is an important skill. Stimulus selection has been studied experimentally, at the single neuron as well as at the population level, by recording from the visual cortex of subjects performing attention-demanding tasks, but studies at the local circuit level are lacking. We conducted simulations of a primary visual cortex (V1) model to provide insight into the local circuit computation underlying stimulus selection in V4. Two small oriented rectangular bars were placed at different locations in the 4 by 4 degree visual field represented by the V1 model, such that they activated different V1 neurons but such that they were both inside the classical receptive field (CRF) of the same V4 neuron. The biased competition framework [Desimone, R., & Duncan, J. (1995). Neural mechanisms of selective visual attention. Annual Review of Neuroscience, 18, 193-222] makes predictions for the response of V4 neurons and the modulation thereof by spatial and feature attention. In our simulation of the V1 network, we obtained results consistent with these predictions for V4 when the model had long-range excitatory projections targeting inhibitory neurons and when spatial attention was mediated by a spatially restricted projection that either inhibited the inhibitory neurons or excited the excitatory neurons. Although it is not clear whether attention effects measured in V4 neurons are generated mostly by local circuits within V4, our simulations suggest that spatial attention at a resolution less than the size of the CRF of a V4 neuron is inherited from upstream areas like V1 and relies on circuits mediating surround suppression at the single neuron level. Furthermore, the model displayed global oscillations in the alpha frequency range (around 10 Hz), whose coherence was highest in the absence of visual stimulation, which is consistent with electroencephalograms recorded in humans. By contrast, when a stimulus was presented the alpha oscillation sped up and became less coherent, whereas at the single column level (40-480 cells) transient beta/gamma oscillations were observed with a frequency between 25 and 50 Hz.

Predicting perception of the wagon wheel illusion

Stroboscopic illumination of a rapidly rotating disk with radial spokes leads to a range of different stationary and moving images as the angular rotation frequency of the disk and the strobe frequency are varied. We compare predictions from the standard correlation model of motion perception with a model based on phase locking observed during periodic stimulation of an integrate-and-fire nonlinear oscillator. The close agreement between theoretical predictions and experimental observations suggests the possibility that periodic forcing of nonlinear neural oscillations may play a role in motion perception.

Two distinct mechanisms shape the reliability of neural responses

Despite intrinsic noise sources, neurons can generate action potentials with remarkable reliability. This reliability is influenced by the characteristics of sensory or synaptic inputs, such as stimulus frequency. Here we use conductance-based models to study the frequency dependence of reliability in terms of the underlying single-cell properties. We are led to distinguish a mean-driven firing regime, where the stimulus mean is sufficient to elicit continuous firing, and a fluctuation-driven firing regime, where spikes are generated by transient stimulus fluctuations. In the mean-driven regime, the stimulus frequency that induces maximum reliability coincides with the firing rate of the cell, whereas in the fluctuation-driven regime, it is determined by the resonance properties of the subthreshold membrane potential. When the stimulus frequency does not match the optimal frequency, the two firing regimes exhibit different "symptoms" of decreased reliability: reduced spike-time precision and reduced spike probability, respectively. As a signature of stochastic resonance, reliable spike generation in the fluctuation-driven regime can benefit from intermediate amounts of noise that boost spike probability without significantly impairing spike-time precision. Our analysis supports the view that neurons are endowed with selection mechanisms that allow only certain stimulus frequencies to induce reliable spiking. By modulating the intrinsic cell properties, the nervous system can thus tune individual neurons to pick out specific input frequency bands with enhanced spike precision or spike probability.

Dynamical phase transitions in periodically driven model neurons

Transitions between dynamical states in integrate-and-fire neuron models with periodic stimuli result from tangent or discontinuous bifurcations of a return map. We study their characteristic scaling laws and show that discontinuous bifurcations exhibit a kind of phase transition intermediate between continuous and first order. In the model-independent spirit of our analysis we show that a six-dimensional (6D) gating variable model with an attracting limit cycle has similar phase transitions, governed by a 1D return map. This reduction to 1D map dynamics should extend to real neurons in a periodic current clamp setting.

Chaotic solutions in the quadratic integrate-and-fire neuron with adaptation

The quadratic integrate-and-fire (QIF) model with adaptation is commonly used as an elementary neuronal model that reproduces the main characteristics of real neurons. In this paper, we introduce a QIF neuron with a nonlinear adaptive current. This model reproduces the neuron-computational features of real neurons and is analytically tractable. It is shown that under a constant current input chaotic firing is possible. In contrast to previous study the neuron is not sinusoidally forced. We show that the spike-triggered adaptation is a key parameter to understand how chaos is generated.

High-order synchronization, transitions, and competition among Arnold tongues in a rotator under harmonic forcing

We consider a rotator whose equation of motion for the angle theta consists of the zeroth and first Fourier modes. Numerical analysis based on the trailing of saddle-node bifurcations is used to locate the n:1 Arnold tongues where synchronization occurs. Several of them are wide enough for high-order synchronization to be seen in passive observations. By sweeping the system parameters within a certain range, we find that the stronger the dependence of theta[over ] on theta , the wider the regions of synchronization. Use of a synchronization index reveals a vast number of very narrow n:m Arnold tongues. A competition phenomenon among the tongues is observed, in that they "push" and "squeeze" one another: as some tongues widen, others narrow. Two mechanisms for transitions between different n:m synchronization states are considered: slow variation of the driving frequency, and the influence of low-frequency noise on the rotator.

Role of interneuron diversity in the cortical microcircuit for attention

Receptive fields of neurons in cortical area V4 are large enough to fit multiple stimuli, making V4 the ideal place to study the effects of selective attention at the single-neuron level. Experiments have revealed evidence for stimulus competition and have characterized the effect thereon of spatial and feature-based attention. We developed a biophysical model with spiking neurons and conductance-based synapses. To account for the comprehensive set of experimental results, it was necessary to include in the model, in addition to regular spiking excitatory (E) cells, two types of interneurons: feedforward interneurons (FFI) and top-down interneurons (TDI). Feature-based attention was mediated by a projection of the TDI to the FFI, stimulus competition was mediated by a cross-columnar excitatory connection to the FFI, whereas spatial attention was mediated by an increase in activity of the feedforward inputs from cortical area V2. The model predicts that spatial attention increases the FFI firing rate, whereas feature-based attention decreases the FFI firing rate and increases the TDI firing rate. During strong stimulus competition, the E cells were synchronous in the beta frequency range (15-35 Hz), but with feature-based attention, they became synchronous in the gamma frequency range (35-50 Hz). We propose that the FFI correspond to fast-spiking, parvalbumin-positive basket cells and that the TDI correspond to cells with a double-bouquet morphology that are immunoreactive to calbindin or calretinin. Taken together, the model results provide an experimentally testable hypothesis for the behavior of two interneuron types under attentional modulation.

Regulation of spike timing in visual cortical circuits

A train of action potentials (a spike train) can carry information in both the average firing rate and the pattern of spikes in the train. But can such a spike-pattern code be supported by cortical circuits? Neurons in vitro produce a spike pattern in response to the injection of a fluctuating current. However, cortical neurons in vivo are modulated by local oscillatory neuronal activity and by top-down inputs. In a cortical circuit, precise spike patterns thus reflect the interaction between internally generated activity and sensory information encoded by input spike trains. We review the evidence for precise and reliable spike timing in the cortex and discuss its computational role.

A review of the integrate-and-fire neuron model: II. Inhomogeneous synaptic input and network properties

The integrate-and-fire neuron model describes the state of a neuron in terms of its membrane potential, which is determined by the synaptic inputs and the injected current that the neuron receives. When the membrane potential reaches a threshold, an action potential (spike) is generated. This review considers the model in which the synaptic input varies periodically and is described by an inhomogeneous Poisson process, with both current and conductance synapses. The focus is on the mathematical methods that allow the output spike distribution to be analyzed, including first passage time methods and the Fokker-Planck equation. Recent interest in the response of neurons to periodic input has in part arisen from the study of stochastic resonance, which is the noise-induced enhancement of the signal-to-noise ratio. Networks of integrate-and-fire neurons behave in a wide variety of ways and have been used to model a variety of neural, physiological, and psychological phenomena. The properties of the integrate-and-fire neuron model with synaptic input described as a temporally homogeneous Poisson process are reviewed in an accompanying paper (Burkitt in Biol Cybern, 2006).

Bifurcation analysis of mode-locking structure in a Hodgkin-Huxley neuron under sinusoidal current

Nervous systems under periodic stimuli display rich dynamical states including mode-locking and chaotic responses, which have been a subject of intense studies in neurodynamics. The bifurcation structure of the Hodgkin-Huxley neuron under sinusoidal stimulus is studied in detail. The mechanisms of the firing onset and rich firing dynamics are studied with the help of the codimension-2 bifurcations, which play the role of the organizing center for myriads of saddle-node, period-doubling, and inverse-flip bifurcations forming the boundaries of the complex mode-locking structure. This study provides a useful insight into the organization of similar bifurcation structures in excitable systems such as neurons under periodic forcing.

Spike Timing Precision and Neural Error Correction: Local Behavior

The effects of spike timing precision and dynamical behavior on error correction in spiking neurons were investigated. Stationary discharges—phase locked, quasiperiodic, or chaotic—were induced in a simulated neuron by presenting pacemaker presynaptic spike trains across a model of a prototypical inhibitory synapse. Reduced timing precision was modeled by jittering presynaptic spike times. Aftereffects of errors—in this communication, missed presynaptic spikes—were determined by comparing postsynaptic spike times between simulations identical except for the presence or absence of errors. Results show that the effects of an error vary greatly depending on the ongoing dynamical behavior. In the case of phase lockings, a high degree of presynaptic spike timing precision can provide significantly faster error recovery. For nonlocked behaviors, isolated missed spikes can have little or no discernible aftereffects (or even serve to paradoxically reduce uncertainty in postsynaptic spike timing), regardless of presynaptic imprecision. This suggests two possible categories of error correction: high-precision locking with rapid recovery and low-precision nonlocked with error immunity.

The possible role of spike patterns in cortical information processing

When the same visual stimulus is presented across many trials, neurons in the visual cortex receive stimulus-related synaptic inputs that are reproducible across trials (S) and inputs that are not (N). The variability of spike trains recorded in the visual cortex and their apparent lack of spike-to-spike correlations beyond that implied by firing rate fluctuations, has been taken as evidence for a low S/N ratio. A recent re-analysis of in vivo cortical data revealed evidence for spike-to-spike correlations in the form of spike patterns. We examine neural dynamics at a higher S/N in order to determine what possible role spike patterns could play in cortical information processing. In vivo-like spike patterns were obtained in model simulations. Superpositions of multiple sinusoidal driving currents were especially effective in producing stable long-lasting patterns. By applying current pulses that were either short and strong or long and weak, neurons could be made to switch from one pattern to another. Cortical neurons with similar stimulus preferences are located near each other, have similar biophysical properties and receive a large number of common synaptic inputs. Hence, recordings of a single neuron across multiple trials are usually interpreted as the response of an ensemble of these neurons during one trial. In the presence of distinct spike patterns across trials there is ambiguity in what would be the corresponding ensemble, it could consist of the same spike pattern for each neuron or a set of patterns across neurons. We found that the spiking response of a neuron receiving these ensemble inputs was determined by the spike-pattern composition, which, in turn, could be modulated dynamically as a means for cortical information processing.

Consistency of nonlinear system response to complex drive signals

The consistency of a nonlinear system's response to a repeated complex waveform drive signal is an important consideration in classical and quantum systems as diverse as lasers, neuronal networks, and manufacturing plants. We show from a consideration of different characteristic waveforms that there is typically an optimal drive amplitude for the most consistent response; internal noise sources dominate for small amplitude driving while deterministic system nonlinearity reduces consistency for large amplitudes. We test this general concept and its measurement experimentally and numerically on the specific example of a laser system.

Inhibitory synchrony as a mechanism for attentional gain modulation

Recordings from area V4 of monkeys have revealed that when the focus of attention is on a visual stimulus within the receptive field of a cortical neuron, two distinct changes can occur: The firing rate of the neuron can change and there can be an increase in the coherence between spikes and the local field potential (LFP) in the gamma-frequency range (30-50 Hz). The hypothesis explored here is that these observed effects of attention could be a consequence of changes in the synchrony of local interneuron networks. We performed computer simulations of a Hodgkin-Huxley type neuron driven by a constant depolarizing current, I, representing visual stimulation and a modulatory inhibitory input representing the effects of attention via local interneuron networks. We observed that the neuron's firing rate and the coherence of its output spike train with the synaptic inputs was modulated by the degree of synchrony of the inhibitory inputs. When inhibitory synchrony increased, the coherence of spiking model neurons with the synaptic input increased, but the firing rate either increased or remained the same. The mean number of synchronous inhibitory inputs was a key determinant of the shape of the firing rate versus current (f-I) curves. For a large number of inhibitory inputs (approximately 50), the f-I curve saturated for large I and an increase in input synchrony resulted in a shift of sensitivity-the model neuron responded to weaker inputs I. For a small number (approximately 10), the f-I curves were non-saturating and an increase in input synchrony led to an increase in the gain of the response-the firing rate in response to the same input was multiplied by an approximately constant factor. The firing rate modulation with inhibitory synchrony was highest when the input network oscillated in the gamma frequency range. Thus, the observed changes in firing rate and coherence of neurons in the visual cortex could be controlled by top-down inputs that regulated the coherence in the activity of a local inhibitory network discharging at gamma frequencies.

Discovering spike patterns in neuronal responses

When a cortical neuron is repeatedly injected with the same fluctuating current stimulus (frozen noise) the timing of the spikes is highly precise from trial to trial and the spike pattern appears to be unique. We show here that the same repeated stimulus can produce more than one reliable temporal pattern of spikes. A new method is introduced to find these patterns in raw multitrial data and is tested on surrogate data sets. Using it, multiple coexisting spike patterns were discovered in pyramidal cells recorded from rat prefrontal cortex in vitro, in data obtained in vivo from the middle temporal area of the monkey (Buracas et al., 1998) and from the cat lateral geniculate nucleus (Reinagel and Reid, 2002). The spike patterns lasted from a few tens of milliseconds in vitro to several seconds in vivo. We conclude that the prestimulus history of a neuron may influence the precise timing of the spikes in response to a stimulus over a wide range of time scales.

Chaos-induced modulation of reliability boosts output firing rate in downstream cortical areas

The reproducibility of neural spike train responses to an identical stimulus across different presentations (trials) has been studied extensively. Reliability, the degree of reproducibility of spike trains, was found to depend in part on the amplitude and frequency content of the stimulus [J. Hunter and J. Milton, J. Neurophysiol. 90, 387 (2003)]. The responses across different trials can sometimes be interpreted as the response of an ensemble of similar neurons to a single stimulus presentation. How does the reliability of the activity of neural ensembles affect information transmission between different cortical areas? We studied a model neural system consisting of two ensembles of neurons with Hodgkin-Huxley-type channels. The first ensemble was driven by an injected sinusoidal current that oscillated in the gamma-frequency range (40 Hz) and its output spike trains in turn drove the second ensemble by fast excitatory synaptic potentials with short term depression. We determined the relationship between the reliability of the first ensemble and the response of the second ensemble. In our paradigm the neurons in the first ensemble were initially in a chaotic state with unreliable and imprecise spike trains. The neurons became entrained to the oscillation and responded reliably when the stimulus power was increased by less than 10%. The firing rate of the first ensemble increased by 30%, whereas that of the second ensemble could increase by an order of magnitude. We also determined the response of the second ensemble when its input spike trains, which had non-Poisson statistics, were replaced by an equivalent ensemble of Poisson spike trains. The resulting output spike trains were significantly different from the original response, as assessed by the metric introduced by Victor and Purpura [J. Neurophysiol. 76, 1310 (1996)]. These results are a proof of principle that weak temporal modulations in the power of gamma-frequency oscillations in a given cortical area can strongly affect firing rate responses downstream by way of reliability in spite of rather modest changes in firing rate in the originating area.

Evaluation of entrainment of a nonlinear neural oscillator to white noise

The Lyapunov exponent for a one-dimensional neural oscillator model, the theta neuron, is computed for white noise forcing, using the steady-state solution to the associated Fokker-Planck equation. The latter is mildly singular, due to the nature of the multiplicative input. In agreement with previous results with similar models, the exponent is negative for all forcing amplitudes, but here it is shown to be small, relative to that for periodic drive, in a range of forcing strengths. Thus the synchronization of an ensemble of independent neurons receiving common but random input can be slow. Moreover, this implies that aperiodic input may be suboptimal, in some contexts, for preserving the reliability of fine spike timing, a potentially important component of the neural "code."

Influence of ionic conductances on spike timing reliability of cortical neurons for suprathreshold rhythmic inputs

Spike timing reliability of neuronal responses depends on the frequency content of the input. We investigate how intrinsic properties of cortical neurons affect spike timing reliability in response to rhythmic inputs of suprathreshold mean. Analyzing reliability of conductance-based cortical model neurons on the basis of a correlation measure, we show two aspects of how ionic conductances influence spike timing reliability. First, they set the preferred frequency for spike timing reliability, which in accordance with the resonance effect of spike timing reliability is well approximated by the firing rate of a neuron in response to the DC component in the input. We demonstrate that a slow potassium current can modulate the spike timing frequency preference over a broad range of frequencies. This result is confirmed experimentally by dynamic-clamp recordings from rat prefrontal cortical neurons in vitro. Second, we provide evidence that ionic conductances also influence spike timing beyond changes in preferred frequency. Cells with the same DC firing rate exhibit more reliable spike timing at the preferred frequency and its harmonics if the slow potassium current is larger and its kinetics are faster, whereas a larger persistent sodium current impairs reliability. We predict that potassium channels are an efficient target for neuromodulators that can tune spike timing reliability to a given rhythmic input.

Amplitude and frequency dependence of spike timing: implications for dynamic regulation

The spike-time reliability of motoneurons in the Aplysia buccal motor ganglion was studied as a function of the frequency content and the relative amplitude of the fluctuations in the neuronal input, calculated as the coefficient of variation (CV). Measurements of spike-time reliability to sinusoidal and aperiodic inputs, as well as simulations of a noisy leaky integrate-and-fire neuron stimulated by spike trains drawn from a periodically modulated process, demonstrate that there are three qualitatively different CV-dependent mechanisms that determine reliability: noise-dominated (CV < 0.05 for Aplysia motoneurons) where spike timing is unreliable regardless of frequency content; resonance-dominated (CV approximately 0.05-0.25) where reliability is reduced by removal of input frequencies equal to motoneuron firing rate; and amplitude-dominated (CV >0.35) where reliability depends on input frequencies greater than motoneuron firing rate. In the resonance-dominated regime, changes in the activity of the presynaptic inhibitory interneuron B4/5 alter motoneuron spike-time reliability. The increases or decreases in reliability occur coincident with small changes in motoneuron spiking rate due to changes in interneuron activity. Injection of a hyperpolarizing current into the motoneuron reproduces the interneuron-induced changes in reliability. The rate-dependent changes in reliability can be understood from the phase-locking properties of regularly spiking motoneurons to periodic inputs. Our observations demonstrate that the ability of a neuron to support a spike-time code can be actively controlled by varying the properties of the neuron and its input.

Reliability of spike timing is a general property of spiking model neurons

The responses of neurons to time-varying injected currents are reproducible on a trial-by-trial basis in vitro, but when a constant current is injected, small variances in interspike intervals across trials add up, eventually leading to a high variance in spike timing. It is unclear whether this difference is due to the nature of the input currents or the intrinsic properties of the neurons. Neuron responses can fail to be reproducible in two ways: dynamical noise can accumulate over time and lead to a desynchronization over trials, or several stable responses can exist, depending on the initial condition. Here we show, through simulations and theoretical considerations, that for a general class of spiking neuron models, which includes, in particular, the leaky integrate-and-fire model as well as nonlinear spiking models, aperiodic currents, contrary to periodic currents, induce reproducible responses, which are stable under noise, change in initial conditions and deterministic perturbations of the input. We provide a theoretical explanation for aperiodic currents that cross the threshold.