Attractor reliability reveals deterministic structure in neuronal spike trains

Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience

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

Finding the event structure of neuronal spike trains

Neurons in sensory systems convey information about physical stimuli in their spike trains. In vitro, single neurons respond precisely and reliably to the repeated injection of the same fluctuating current, producing regions of elevated firing rate, termed events. Analysis of these spike trains reveals that multiple distinct spike patterns can be identified as trial-to-trial correlations between spike times (Fellous, Tiesinga, Thomas, & Sejnowski, 2004 ). Finding events in data with realistic spiking statistics is challenging because events belonging to different spike patterns may overlap. We propose a method for finding spiking events that uses contextual information to disambiguate which pattern a trial belongs to. The procedure can be applied to spike trains of the same neuron across multiple trials to detect and separate responses obtained during different brain states. The procedure can also be applied to spike trains from multiple simultaneously recorded neurons in order to identify volleys of near-synchronous activity or to distinguish between excitatory and inhibitory neurons. The procedure was tested using artificial data as well as recordings in vitro in response to fluctuating current waveforms.

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.

Using interspike intervals to quantify noise effects on spike trains in temperature encoding neurons

This paper examines how noise interacts with the non-linear dynamical mechanisms of neuronal stimulus. We study the spike trains generated by a minimal Hodgkin-Huxley type model of a cold receptor neuron. The distributions of interspike intervals(ISIs) of purely deterministic simulations exhibit considerable differences compared to the noisy ones. We quantify the effect of noise using ISI return plots and the ISI-distance recently proposed by Kreuz et al. (J Neurosci Meth, 165:151-161, 2007). It is shown that the spike trains of a cold receptor neuron are more strongly affected by noise for low temperatures than for high temperatures. This trend is also observed in both regimes of cold receptors: tonic firing(which occurs for low and high temperatures) and bursting (which occurs for intermediate temperatures).

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.

Measuring spike train synchrony

Estimating the degree of synchrony or reliability between two or more spike trains is a frequent task in both experimental and computational neuroscience. In recent years, many different methods have been proposed that typically compare the timing of spikes on a certain time scale to be optimized by the analyst. Here, we propose the ISI-distance, a simple complementary approach that extracts information from the interspike intervals by evaluating the ratio of the instantaneous firing rates. The method is parameter free, time scale independent and easy to visualize as illustrated by an application to real neuronal spike trains obtained in vitro from rat slices. In a comparison with existing approaches on spike trains extracted from a simulated Hindemarsh-Rose network, the ISI-distance performs as well as the best time-scale-optimized measure based on spike timing.

Synaptic input statistics tune the variability and reproducibility of neuronal responses

Synaptic waveforms, constructed from excitatory and inhibitory presynaptic Poisson trains, are presented to living and computational neurons. We review how the average output of a neuron (e.g., the firing rate) is set by the difference between excitatory and inhibitory event rates while neuronal variability is set by their sum. We distinguish neuronal variability from reproducibility. Variability quantifies how much an output measure is expected to vary; for example, the interspike interval coefficient of variation quantifies the typical range of interspike intervals. Reproducibility quantifies the similarity of neuronal outputs in response to repeated presentations of identical stimuli. Although variability and reproducibility are conceptually distinct, we show that, for ideal current source synapses, reproducibility is defined entirely by variability. For physiologically realistic conductance-based synapses, however, reproducibility is distinct from variability and average output, set by the Poisson rate and the degree of synchrony within the synaptic waveform.

Dynamical basis of irregular spiking in NMDA-driven prefrontal cortex neurons

Slow N-Methyl-D-aspartic acid (NMDA) synaptic currents are assumed to strongly contribute to the persistently elevated firing rates observed in prefrontal cortex (PFC) during working memory. During persistent activity, spiking of many neurons is highly irregular. Here we report that highly irregular firing can be induced through a combination of NMDA- and dopamine D1 receptor agonists applied to adult PFC neurons in vitro. The highest interspike-interval (ISI) variability occurred in a transition regime where the subthreshold membrane potential distribution shifts from mono- to bimodality, while neurons with clearly mono- or bimodal distributions fired much more regularly. Predictability within irregular ISI series was significantly higher than expected from a noise-driven linear process, indicating that it might best be described through complex (potentially chaotic) nonlinear deterministic processes. Accordingly, the phenomena observed in vitro could be reproduced in purely deterministic biophysical model neurons. High spiking irregularity in these models emerged within a chaotic, close-to-bifurcation regime characterized by a shift of the membrane potential distribution from mono- to bimodality and by similar ISI return maps as observed in vitro. The nonlinearity of NMDA conductances was crucial for inducing this regime. NMDA-induced irregular dynamics may have important implications for computational processes during working memory and neural coding.

Information Geometry of Interspike Intervals in Spiking Neurons

An information geometrical method is developed for characterizing or classifying neurons in cortical areas, whose spike rates fluctuate in time. Under the assumption that the interspike intervals of a spike sequence of a neuron obey a gamma process with a time-variant spike rate and a fixed shape parameter, we formulate the problem of characterization as a semiparametric statistical estimation, where the spike rate is a nuisance parameter. We derive optimal criteria from the information geometrical viewpoint when certain assumptions are added to the formulation, and we show that some existing measures, such as the coefficient of variation and the local variation, are expressed as estimators of certain functions under the same assumptions.

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.

Increased noise level of purkinje cell activities minimizes impact of their modulation during sensorimotor control

While firing rate is well established as a relevant parameter for encoding information exchanged between neurons, the significance of other parameters is more conjectural. Here, we show that regularity of neuronal spike activities affects sensorimotor processing in tottering mutants, which suffer from a mutation in P/Q-type voltage-gated calcium channels. While the modulation amplitude of the simple spike firing rate of their floccular Purkinje cells during optokinetic stimulation is indistinguishable from that of wild-types, the regularity of their firing is markedly disrupted. The gain and phase values of tottering's compensatory eye movements are indistinguishable from those of flocculectomized wild-types or from totterings with the flocculus treated with P/Q-type calcium channel blockers. Moreover, normal eye movements can be evoked in tottering when the flocculus is electrically stimulated with regular spike trains mimicking the firing pattern of normal simple spikes. This study demonstrates the importance of regularity of firing in Purkinje cells for neuronal information processing.

Masking and scrambling in the auditory thalamus of awake rats by Gaussian and modulated noises

This paper provides a look at how modulated broad-band noises modulate the thalamic response evoked by brief probe sounds in the awake animal. We demonstrate that noise not only attenuates the response to probe sounds (masking) but also changes the temporal response pattern (scrambling). Two brief probe sounds, a Gaussian noise burst and a brief sinusoidal tone, were presented in silence and in three ongoing noises. The three noises were targeted at activating the auditory system in qualitatively distinct ways. Dynamic ripple noise, containing many random tone-like elements, is targeted at those parts of the auditory system that respond well to tones. International Collegium of Rehabilitative Audiology noise, comprised of the sum of several simultaneous streams of Schroeder-phase speech, is targeted at those parts of the auditory system that respond well to modulated sounds but lack a well defined response to tones. Gaussian noise is targeted at those parts of the auditory system that respond to acoustic energy regardless of modulation. All noises both attenuated and decreased the precise temporal repeatability of the onset response to probe sounds. In addition, the modulated noises induced context-specific changes in the temporal pattern of the response to probe sounds. Scrambling of the temporal response pattern may be a direct neural correlate of the unfortunate experience of being able to hear, but not understand, speech sounds in noisy environments.

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.

Differences in Spiking Patterns Among Cortical Neurons

Spike sequences recorded from four cortical areas of an awake behaving monkey were examined to explore characteristics that vary among neurons. We found that a measure of the local variation of interspike intervals, LV, is nearly the same for every spike sequence for any given neuron, while it varies significantly among neurons. The distributions of LV values for neuron ensembles in three of the four areas were found to be distinctly bimodal. Two groups of neurons classified according to the spiking irregularity exhibit different responses to the same stimulus. This suggests that neurons in each area can be classified into different groups possessing unique spiking statistics and corresponding functional properties.

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.

A new correlation-based measure of spike timing reliability

We introduce a new correlation-based measure of spike timing reliability. Unlike other measures, it does not require the definition of a posteriori "events". It relies on only one parameter, which relates to the timescale of spike timing precision. We test the measure on surrogate data sets with varying amounts of spike time jitter, and missing or additional spikes, and compare it with a widely used histogram-based measure. The measure is efficient and faithful in characterizing spike timing reliability and produces smaller errors in the reliability estimate than the histogram-based measure based on the same number of trials.

Effects of noise on the spike timing precision of retinal ganglion cells

Information in a spike train is limited by variability in the spike timing. This variability is caused by noise from several sources including synapses and membrane channels; but how deleterious each noise source is and how they affect spike train coding is unknown. Combining physiology and a multicompartment model, we studied the effect of synaptic input noise and voltage-gated channel noise on spike train reliability for a mammalian ganglion cell. For tonic stimuli, the SD of the interspike intervals increased supralinearly with increasing interspike interval. When the cell was driven by current injection, voltage-gated channel noise and background synaptic noise caused fluctuations in the interspike interval of comparable amplitude. Spikes initiated on the dendrites could cause additional spike timing fluctuations. For transient stimuli, synaptic noise was dominant and spontaneous background activity strongly increased fluctuations in spike timing but decreased the latency of the first spike.

The effects of input rate and synchrony on a coincidence detector: analytical solution

We derive analytically the solution for the output rate of the ideal coincidence detector. The solution is for an arbitrary number of input spike trains with identical binomial count distributions (which includes Poisson statistics as a special case) and identical arbitrary pairwise cross-correlations, from zero correlation (independent processes) to complete correlation (identical processes).

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.