Spike initiation by transmembrane current: a white-noise analysis

The neuron as a direct data-driven controller

In the quest to model neuronal function amid gaps in physiological data, a promising strategy is to develop a normative theory that interprets neuronal physiology as optimizing a computational objective. This study extends current normative models, which primarily optimize prediction, by conceptualizing neurons as optimal feedback controllers. We posit that neurons, especially those beyond early sensory areas, steer their environment toward a specific desired state through their output. This environment comprises both synaptically interlinked neurons and external motor sensory feedback loops, enabling neurons to evaluate the effectiveness of their control via synaptic feedback. To model neurons as biologically feasible controllers which implicitly identify loop dynamics, infer latent states, and optimize control we utilize the contemporary direct data-driven control (DD-DC) framework. Our DD-DC neuron model explains various neurophysiological phenomena: the shift from potentiation to depression in spike-timing-dependent plasticity with its asymmetry, the duration and adaptive nature of feedforward and feedback neuronal filters, the imprecision in spike generation under constant stimulation, and the characteristic operational variability and noise in the brain. Our model presents a significant departure from the traditional, feedforward, instant-response McCulloch-Pitts-Rosenblatt neuron, offering a modern, biologically informed fundamental unit for constructing neural networks.

Stimulus-dependent differences in cortical versus subcortical contributions to visual detection in mice

The primary visual cortex (V1) and the superior colliculus (SC) both occupy stations early in the processing of visual information. They have long been thought to perform distinct functions, with the V1 supporting the perception of visual features and the SC regulating orienting to visual inputs. However, growing evidence suggests that the SC supports the perception of many of the same visual features traditionally associated with the V1. To distinguish V1 and SC contributions to visual processing, it is critical to determine whether both areas causally contribute to the detection of specific visual stimuli. Here, mice reported changes in visual contrast or luminance near their perceptual threshold while white noise patterns of optogenetic stimulation were delivered to V1 or SC inhibitory neurons. We then performed a reverse correlation analysis on the optogenetic stimuli to estimate a neuronal-behavioral kernel (NBK), a moment-to-moment estimate of the impact of V1 or SC inhibition on stimulus detection. We show that the earliest moments of stimulus-evoked activity in the SC are critical for the detection of both luminance and contrast changes. Strikingly, there was a robust stimulus-aligned modulation in the V1 contrast-detection NBK but no sign of a comparable modulation for luminance detection. The data suggest that behavioral detection of visual contrast depends on both V1 and SC spiking, whereas mice preferentially use SC activity to detect changes in luminance. Electrophysiological recordings showed that neurons in both the SC and V1 responded strongly to both visual stimulus types, while the reverse correlation analysis reveals when these neuronal signals actually contribute to visually guided behaviors.

Simulations predict differing phase responses to excitation vs. inhibition in theta-resonant pyramidal neurons

Rhythmic activity is ubiquitous in neural systems, with theta-resonant pyramidal neurons integrating rhythmic inputs in many cortical structures. Impedance analysis has been widely used to examine frequency-dependent responses of neuronal membranes to rhythmic inputs, but it assumes that the neuronal membrane is a linear system, requiring the use of small signals to stay in a near-linear regime. However, postsynaptic potentials are often large and trigger nonlinear mechanisms (voltage-gated ion channels). The goals of this work were to 1) develop an analysis method to evaluate membrane responses in this nonlinear domain and 2) explore phase relationships between rhythmic stimuli and subthreshold and spiking membrane potential (Vmemb) responses in models of theta-resonant pyramidal neurons. Responses in these output regimes were asymmetrical, with different phase shifts during hyperpolarizing and depolarizing half-cycles. Suprathreshold theta-rhythmic stimuli produced nonstationary Vmemb responses. Sinusoidal inputs produced "phase retreat": action potentials occurred progressively later in cycles of the input stimulus, resulting from adaptation. Sinusoidal current with increasing amplitude over cycles produced "phase advance": action potentials occurred progressively earlier. Phase retreat, phase advance, and subthreshold phase shifts were modulated by multiple ion channel conductances. Our results suggest differential responses of cortical neurons depending on the frequency of oscillatory input, which will play a role in neuronal responses to shifts in network state. We hypothesize that intrinsic cellular properties complement network properties and contribute to in vivo phase-shift phenomena such as phase precession, seen in place and grid cells, and phase roll, also observed in hippocampal CA1 neurons.NEW & NOTEWORTHY We augmented electrical impedance analysis to characterize phase shifts between large-amplitude current stimuli and nonlinear, asymmetric membrane potential responses. We predict different frequency-dependent phase shifts in response excitation vs. inhibition, as well as shifts in spike timing over multiple input cycles, in theta-resonant pyramidal neurons. We hypothesize that these effects contribute to navigation-related phenomena such as phase precession and phase roll. Our neuron-level hypothesis complements, rather than falsifies, prior network-level explanations of these phenomena.

Parameter and coupling estimation in small networks of Izhikevich's neurons

Nowadays, experimental techniques allow scientists to have access to large amounts of data. In order to obtain reliable information from the complex systems that produce these data, appropriate analysis tools are needed. The Kalman filter is a frequently used technique to infer, assuming a model of the system, the parameters of the model from uncertain observations. A well-known implementation of the Kalman filter, the unscented Kalman filter (UKF), was recently shown to be able to infer the connectivity of a set of coupled chaotic oscillators. In this work, we test whether the UKF can also reconstruct the connectivity of small groups of coupled neurons when their links are either electrical or chemical synapses. In particular, we consider Izhikevich neurons and aim to infer which neurons influence each other, considering simulated spike trains as the experimental observations used by the UKF. First, we verify that the UKF can recover the parameters of a single neuron, even when the parameters vary in time. Second, we analyze small neural ensembles and demonstrate that the UKF allows inferring the connectivity between the neurons, even for heterogeneous, directed, and temporally evolving networks. Our results show that time-dependent parameter and coupling estimation is possible in this nonlinearly coupled system.

Precise Spiking Motifs in Neurobiological and Neuromorphic Data

Why do neurons communicate through spikes? By definition, spikes are all-or-none neural events which occur at continuous times. In other words, spikes are on one side binary, existing or not without further details, and on the other, can occur at any asynchronous time, without the need for a centralized clock. This stands in stark contrast to the analog representation of values and the discretized timing classically used in digital processing and at the base of modern-day neural networks. As neural systems almost systematically use this so-called event-based representation in the living world, a better understanding of this phenomenon remains a fundamental challenge in neurobiology in order to better interpret the profusion of recorded data. With the growing need for intelligent embedded systems, it also emerges as a new computing paradigm to enable the efficient operation of a new class of sensors and event-based computers, called neuromorphic, which could enable significant gains in computation time and energy consumption-a major societal issue in the era of the digital economy and global warming. In this review paper, we provide evidence from biology, theory and engineering that the precise timing of spikes plays a crucial role in our understanding of the efficiency of neural networks.

Coexisting neuronal coding strategies in the barrel cortex

During tactile sensation by rodents, whisker movements across surfaces generate complex whisker motions, including discrete, transient stick-slip events, which carry information about surface properties. The characteristics of these events and how the brain encodes this tactile information remain enigmatic. We found that cortical neurons show a mixture of synchronized and nontemporally correlated spikes in their tactile responses. Synchronous spikes convey the magnitude of stick-slip events by numerous aspects of temporal coding. These spikes show preferential selectivity for kinetic and kinematic whisker motion. By contrast, asynchronous spikes in each neuron convey the magnitude of stick-slip events by their discharge rates, response probability, and interspike intervals. We further show that the differentiation between these two types of activity is highly dependent on the magnitude of stick-slip events and stimulus and response history. These results suggest that cortical neurons transmit multiple components of tactile information through numerous coding strategies.

Error correction and improved precision of spike timing in converging cortical networks

The brain propagates neuronal signals accurately and rapidly. Nevertheless, whether and how a pool of cortical neurons transmits an undistorted message to a target remains unclear. We apply optogenetic white noise signals to small assemblies of cortical pyramidal cells (PYRs) in freely moving mice. The directly activated PYRs exhibit a spike timing precision of several milliseconds. Instead of losing precision, interneurons driven via synaptic activation exhibit higher precision with respect to the white noise signal. Compared with directly activated PYRs, postsynaptic interneuron spike trains allow better signal reconstruction, demonstrating error correction. Data-driven modeling shows that nonlinear amplification of coincident spikes can generate error correction and improved precision. Over multiple applications of the same signal, postsynaptic interneuron spiking is most reliable at timescales ten times shorter than those of the presynaptic PYR, exhibiting temporal coding. Similar results are observed in hippocampal region CA1. Coincidence detection of convergent inputs enables messages to be precisely propagated between cortical PYRs and interneurons.

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PYR-to-interneuron spike transmission exhibits error correction and improved precision

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Interneuron precision is higher when a larger pool of presynaptic PYRs is recruited

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Error correction and improved precision are consistent with coincidence detection

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Interneurons activated by synaptic transmission act as temporal coders

Physiological noise facilitates multiplexed coding of vibrotactile-like signals in somatosensory cortex

Neurons can use different aspects of their spiking to simultaneously represent (multiplex) different features of a stimulus. For example, some pyramidal neurons in primary somatosensory cortex (S1) use the rate and timing of their spikes to, respectively, encode the intensity and frequency of vibrotactile stimuli. Doing so has several requirements. Because they fire at low rates, pyramidal neurons cannot entrain 1:1 with high-frequency (100 to 600 Hz) inputs and, instead, must skip (i.e., not respond to) some stimulus cycles. The proportion of skipped cycles must vary inversely with stimulus intensity for firing rate to encode stimulus intensity. Spikes must phase-lock to the stimulus for spike times (intervals) to encode stimulus frequency, but, in addition, skipping must occur irregularly to avoid aliasing. Using simulations and in vitro experiments in which mouse S1 pyramidal neurons were stimulated with inputs emulating those induced by vibrotactile stimuli, we show that fewer cycles are skipped as stimulus intensity increases, as required for rate coding, and that intrinsic or synaptic noise can induce irregular skipping without disrupting phase locking, as required for temporal coding. This occurs because noise can modulate the reliability without disrupting the precision of spikes evoked by small-amplitude, fast-onset signals. Specifically, in the fluctuation-driven regime associated with sparse spiking, rate and temporal coding are both paradoxically improved by the strong synaptic noise characteristic of the intact cortex. Our results demonstrate that multiplexed coding by S1 pyramidal neurons is not only feasible under in vivo conditions, but that background synaptic noise is actually beneficial.

SNR Enhancement of Direct Absorption Spectroscopy Utilizing an Improved Particle Swarm Algorithm

A noise elimination method based on an improved particle swarm algorithm is applied to direct absorption spectroscopy. The algorithm combines the theory of spectral line shape to calculate a fitness function according to the original spectra. Comparing the particles and the fitness function to calculate the updating direction, and position of particles, the iterative update finally finds the optimal solution. The algorithm is applied to direct absorption spectroscopy to measure methane; compared with the signal without algorithm processing, the signal-to-noise ratio (SNR) is improved by 4.17 times, and the minimum detection limit in the experiment is 15.3 ppb. R2 = 0.9999 is calculated in the calibration experiment, and the error is less than 0.1 ppm in the repeatability experiment of constant methane at 2 ppm concentration.

Perceptual Weighting of V1 Spikes Revealed by Optogenetic White Noise Stimulation

During visually guided behaviors, mere hundreds of milliseconds can elapse between a sensory input and its associated behavioral response. How spikes occurring at different times are integrated to drive perception and action remains poorly understood. We delivered random trains of optogenetic stimulation (white noise) to excite inhibitory interneurons in V1 of mice of both sexes while they performed a visual detection task. We then performed a reverse correlation analysis on the optogenetic stimuli to generate a neuronal-behavioral kernel, an unbiased, temporally precise estimate of how suppression of V1 spiking at different moments around the onset of a visual stimulus affects detection of that stimulus. Electrophysiological recordings enabled us to capture the effects of optogenetic stimuli on V1 responsivity and revealed that the earliest stimulus-evoked spikes are preferentially weighted for guiding behavior. These data demonstrate that white noise optogenetic stimulation is a powerful tool for understanding how patterns of spiking in neuronal populations are decoded in generating perception and action.SIGNIFICANCE STATEMENT During visually guided actions, continuous chains of neurons connect our retinas to our motoneurons. To unravel circuit contributions to behavior, it is crucial to establish the relative functional position(s) that different neural structures occupy in processing and relaying the signals that support rapid, precise responses. To address this question, we randomly inhibited activity in mouse V1 throughout the stimulus-response cycle while the animals did many repetitions of a visual task. The period that led to impaired performance corresponded to the earliest stimulus-driven response in V1, with no effect of inhibition immediately before or during late stages of the stimulus-driven response. This approach offers experimenters a powerful method for uncovering the temporal weighting of spikes from stimulus to response.

Neuronal avalanches in Watts-Strogatz networks of stochastic spiking neurons

Networks of stochastic leaky integrate-and-fire neurons, both at the mean-field level and in square lattices, present a continuous absorbing phase transition with power-law neuronal avalanches at the critical point. Here we complement these results showing that small-world Watts-Strogatz networks have mean-field critical exponents for any rewiring probability p>0. For the ring (p=0), the exponents are the same from the dimension d=1 of the directed-percolation class. In the model, firings are stochastic and occur in discrete time steps, based on a sigmoidal firing probability function. Each neuron has a membrane potential that integrates the signals received from its neighbors. The membrane potentials are subject to a leakage parameter. We study topologies with a varied number of neuron connections and different values of the leakage parameter. Results indicate that the dynamic range is larger for p=0. We also study a homeostatic synaptic depression mechanism to self-organize the network towards the critical region. These stochastic oscillations are characteristic of the so-called self-organized quasicriticality.

Identification of Multiple Noise Sources Improves Estimation of Neural Responses across Stimulus Conditions

Most models of neural responses are constructed to reproduce the average response to inputs but lack the flexibility to capture observed variability in responses. The origins and structure of this variability have significant implications for how information is encoded and processed in the nervous system, both by limiting information that can be conveyed and by determining processing strategies that are favorable for minimizing its negative effects. Here, we present a new modeling framework that incorporates multiple sources of noise to better capture observed features of neural response variability across stimulus conditions. We apply this model to retinal ganglion cells at two different ambient light levels and demonstrate that it captures the full distribution of responses. Further, the model reveals light level-dependent changes that could not be seen with previous models, showing both large changes in rectification of nonlinear circuit elements and systematic differences in the contributions of different noise sources under different conditions.

Cortical gamma-band resonance preferentially transmits coherent input

Synchronization has been implicated in neuronal communication, but causal evidence remains indirect. We use optogenetics to generate depolarizing currents in pyramidal neurons of the cat visual cortex, emulating excitatory synaptic inputs under precise temporal control, while measuring spike output. The cortex transforms constant excitation into strong gamma-band synchronization, revealing the well-known cortical resonance. Increasing excitation with ramps increases the strength and frequency of synchronization. Slow, symmetric excitation profiles reveal hysteresis of power and frequency. White-noise input sequences enable causal analysis of network transmission, establishing that the cortical gamma-band resonance preferentially transmits coherent input components. Models composed of recurrently coupled excitatory and inhibitory units uncover a crucial role of feedback inhibition and suggest that hysteresis can arise through spike-frequency adaptation. The presented approach provides a powerful means to investigate the resonance properties of local circuits and probe how these properties transform input and shape transmission.

Rhythmic synchronization has been implicated in neuronal communication, yet causal evidence has remained scarce. Lewis et al. optogenetically stimulate the visual cortex to emulate synaptic input while recording spike output. Cortex resonates at the gamma band (30–90 Hz) and preferentially transmits input that is coherent to the ongoing gamma-band rhythm.

Time- and frequency-resolved covariance analysis for detection and characterization of seizures from intracraneal EEG recordings

The amount of power in different frequency bands of the electroencephalogram (EEG) carries information about the behavioral state of a subject. Hence, neurologists treating epileptic patients monitor the temporal evolution of the different bands. We propose a covariance-based method to detect and characterize epileptic seizures operating on the band-filtered EEG signal. The algorithm is unsupervised and performs a principal component analysis of intra-cranial EEG recordings, detecting transient fluctuations of the power in each frequency band. Its simplicity makes it suitable for online implementation. Good sampling of the non-ictal periods is required, while no demands are imposed on the amount of data during ictal activity. We tested the method with 32 seizures registered in 5 patients. The area below the resulting receiver-operating characteristic curves was 87% for the detection of seizures and 91% for the detection of recruited electrodes. To identify the behaviorally relevant correlates of the physiological signal, we identified transient changes in the variance of each band that were correlated with the degree of loss of consciousness, the latter assessed by the so-called Consciousness Seizure Scale, summarizing the performance of the subject in a number of behavioral tests requested during seizures. We concluded that those crisis with maximal impairment of consciousness tended to exhibit an increase in variance approximately 40 s after seizure onset, with predominant power in the theta and alpha bands and reduced delta and beta activity.

Variety Is the Spice of Life: Positive and Negative Effects of Noise in Electrical Stimulation of the Nervous System

Noisy stimuli may hold the key for optimal electrical stimulation of the nervous system. Possible mechanisms of noise’s impact upon neuronal function are discussed, including intracellular, extracellular, and systems-level mechanisms. Specifically, channel resonance, stochastic resonance, high conductance states, and network binding are investigated. These mechanisms are examined and possible directions of growth for the field are discussed, with examples of applications provided from the fields of deep brain stimulation or spinal cord injury. Together, this review highlights the theoretical basis and evidence base for the use of noise to enhance current stimulation paradigms of the nervous system.

Pre-Synaptic Pool Modification (PSPM): A supervised learning procedure for recurrent spiking neural networks

Learning synaptic weights of spiking neural network (SNN) models that can reproduce target spike trains from provided neural firing data is a central problem in computational neuroscience and spike-based computing. The discovery of the optimal weight values can be posed as a supervised learning task wherein the weights of the model network are chosen to maximize the similarity between the target spike trains and the model outputs. It is still largely unknown whether optimizing spike train similarity of highly recurrent SNNs produces weight matrices similar to those of the ground truth model. To this end, we propose flexible heuristic supervised learning rules, termed Pre-Synaptic Pool Modification (PSPM), that rely on stochastic weight updates in order to produce spikes within a short window of the desired times and eliminate spikes outside of this window. PSPM improves spike train similarity for all-to-all SNNs and makes no assumption about the post-synaptic potential of the neurons or the structure of the network since no gradients are required. We test whether optimizing for spike train similarity entails the discovery of accurate weights and explore the relative contributions of local and homeostatic weight updates. Although PSPM improves similarity between spike trains, the learned weights often differ from the weights of the ground truth model, implying that connectome inference from spike data may require additional constraints on connectivity statistics. We also find that spike train similarity is sensitive to local updates, but other measures of network activity such as avalanche distributions, can be learned through synaptic homeostasis.

Deterministic networks for probabilistic computing

Neuronal network models of high-level brain functions such as memory recall and reasoning often rely on the presence of some form of noise. The majority of these models assumes that each neuron in the functional network is equipped with its own private source of randomness, often in the form of uncorrelated external noise. In vivo, synaptic background input has been suggested to serve as the main source of noise in biological neuronal networks. However, the finiteness of the number of such noise sources constitutes a challenge to this idea. Here, we show that shared-noise correlations resulting from a finite number of independent noise sources can substantially impair the performance of stochastic network models. We demonstrate that this problem is naturally overcome by replacing the ensemble of independent noise sources by a deterministic recurrent neuronal network. By virtue of inhibitory feedback, such networks can generate small residual spatial correlations in their activity which, counter to intuition, suppress the detrimental effect of shared input. We exploit this mechanism to show that a single recurrent network of a few hundred neurons can serve as a natural noise source for a large ensemble of functional networks performing probabilistic computations, each comprising thousands of units.

Temporal Coding of Reward Value in Monkey Ventral Striatal Tonically Active Neurons

The rostromedioventral striatum is critical for behavior dependent on evaluating rewards. We asked what contribution tonically active neurons (TANs), the putative striatal cholinergic interneurons, make in coding reward value in this part of the striatum. Two female monkeys were given the option to accept or reject an offered reward in each trial, the value of which was signaled by a visual cue. Forty-five percent of the TANs use temporally modulated activity to encode information about discounted value. These responses were significantly better represented using principal component analysis than by just counting spikes. The temporal coding is straightforward: the spikes are distributed according to a sinusoidal envelope of activity that changes gain, ranging from positive to negative according to discounted value. Our results show that the information about the relative value of an offered reward is temporally encoded in neural spike trains of TANs. This temporal coding may allow well tuned, coordinated behavior to emerge.SIGNIFICANCE STATEMENT Ever since the discovery that neurons use trains of pulses to transmit information, it seemed self-evident that information would be encoded into the pattern of the spikes. However, there is not much evidence that spike patterns encode cognitive information. We find that a set of interneurons, the tonically active neurons (TANs) in monkeys' striatum, use temporal patterns of response to encode information about the discounted value of offered rewards. The code seems straightforward: a sinusoidal envelope that changes gain according to the discounted value of the offer, describes the rate of spiking across time. This temporal modulation may provide a means to synchronize these interneurons and the activity of other neural elements including principal output neurons.

An axon initial segment is required for temporal precision in action potential encoding by neuronal populations

Central neurons initiate action potentials (APs) in the axon initial segment (AIS), a compartment characterized by a high concentration of voltage-dependent ion channels and specialized cytoskeletal anchoring proteins arranged in a regular nanoscale pattern. Although the AIS was a key evolutionary innovation in neurons, the functional benefits it confers are not clear. Using a mutation of the AIS cytoskeletal protein βIV-spectrin, we here establish an in vitro model of neurons with a perturbed AIS architecture that retains nanoscale order but loses the ability to maintain a high Na_V density. Combining experiments and simulations, we show that a high Na_V density in the AIS is not required for axonal AP initiation; it is, however, crucial for a high bandwidth of information encoding and AP timing precision. Our results provide the first experimental demonstration of axonal AP initiation without high axonal channel density and suggest that increasing the bandwidth of the neuronal code and, hence, the computational efficiency of network function, was a major benefit of the evolution of the AIS.

Nonlinear Modeling of Neural Interaction for Spike Prediction Using the Staged Point-Process Model

Neurons communicate nonlinearly through spike activities. Generalized linear models (GLMs) describe spike activities with a cascade of a linear combination across inputs, a static nonlinear function, and an inhomogeneous Bernoulli or Poisson process, or Cox process if a self-history term is considered. This structure considers the output nonlinearity in spike generation but excludes the nonlinear interaction among input neurons. Recent studies extend GLMs by modeling the interaction among input neurons with a quadratic function, which considers the interaction between every pair of input spikes. However, quadratic effects may not fully capture the nonlinear nature of input interaction. We therefore propose a staged point-process model to describe the nonlinear interaction among inputs using a few hidden units, which follows the idea of artificial neural networks. The output firing probability conditioned on inputs is formed as a cascade of two linear-nonlinear (a linear combination plus a static nonlinear function) stages and an inhomogeneous Bernoulli process. Parameters of this model are estimated by maximizing the log likelihood on output spike trains. Unlike the iterative reweighted least squares algorithm used in GLMs, where the performance is guaranteed by the concave condition, we propose a modified Levenberg-Marquardt (L-M) algorithm, which directly calculates the Hessian matrix of the log likelihood, for the nonlinear optimization in our model. The proposed model is tested on both synthetic data and real spike train data recorded from the dorsal premotor cortex and primary motor cortex of a monkey performing a center-out task. Performances are evaluated by discrete-time rescaled Kolmogorov-Smirnov tests, where our model statistically outperforms a GLM and its quadratic extension, with a higher goodness-of-fit in the prediction results. In addition, the staged point-process model describes nonlinear interaction among input neurons with fewer parameters than quadratic models, and the modified L-M algorithm also demonstrates fast convergence.

Attractor dynamics of a Boolean model of a brain circuit controlled by multiple parameters

Studies of Boolean recurrent neural networks are briefly introduced with an emphasis on the attractor dynamics determined by the sequence of distinct attractors observed in the limit cycles. We apply this framework to a simplified model of the basal ganglia-thalamocortical circuit where each brain area is represented by a "neuronal" node in a directed graph. Control parameters ranging from neuronal excitability that affects all cells to targeted local connections modified by a new adaptive plasticity rule, and the regulation of the interactive feedback affecting the external input stream of information, allow the network dynamics to switch between stable domains delimited by highly discontinuous boundaries and reach very high levels of complexity with specific configurations. The significance of this approach with regard to brain circuit studies is briefly discussed.

Differential polarization of cortical pyramidal neuron dendrites through weak extracellular fields

The rise of transcranial current stimulation (tCS) techniques have sparked an increasing interest in the effects of weak extracellular electric fields on neural activity. These fields modulate ongoing neural activity through polarization of the neuronal membrane. While the somatic polarization has been investigated experimentally, the frequency-dependent polarization of the dendritic trees in the presence of alternating (AC) fields has received little attention yet. Using a biophysically detailed model with experimentally constrained active conductances, we analyze the subthreshold response of cortical pyramidal cells to weak AC fields, as induced during tCS. We observe a strong frequency resonance around 10-20 Hz in the apical dendrites sensitivity to polarize in response to electric fields but not in the basal dendrites nor the soma. To disentangle the relative roles of the cell morphology and active and passive membrane properties in this resonance, we perform a thorough analysis using simplified models, e.g. a passive pyramidal neuron model, simple passive cables and reconstructed cell model with simplified ion channels. We attribute the origin of the resonance in the apical dendrites to (i) a locally increased sensitivity due to the morphology and to (ii) the high density of h-type channels. Our systematic study provides an improved understanding of the subthreshold response of cortical cells to weak electric fields and, importantly, allows for an improved design of tCS stimuli.

Theta-frequency selectivity in the somatic spike-triggered average of rat hippocampal pyramidal neurons is dependent on HCN channels

The ability to distill specific frequencies from complex spatiotemporal patterns of afferent inputs is a pivotal functional requirement for neurons residing in networks receiving frequency-multiplexed inputs. Although the expression of theta-frequency subthreshold resonance is established in hippocampal pyramidal neurons, it is not known if their spike initiation dynamics manifest spectral selectivity, or if their intrinsic properties are tuned to process gamma-frequency inputs. Here, we measured the spike-triggered average (STA) of rat hippocampal pyramidal neurons through electrophysiological recordings and quantified spectral selectivity in their spike initiation dynamics and their coincidence detection window (CDW). Our results revealed strong theta-frequency selectivity in the STA, which was also endowed with gamma-range CDW, with prominent neuron-to-neuron variability that manifested distinct pairwise dissociations and correlations with different intrinsic measurements. Furthermore, we demonstrate that the STA and its measurements substantially adapted to the state of the neuron defined by its membrane potential and to the statistics of its afferent inputs. Finally, we tested the effect of pharmacologically blocking the hyperpolarization-activated cyclic-nucleotide-gated (HCN) channels on the STA and found that the STA characteristic frequency reduced significantly to the delta-frequency band after HCN channel blockade. This delta-frequency selectivity in the STA emerged in the absence of subthreshold resonance, which was abolished by HCN channel blockade, thereby confirming computational predictions on the dissociation between these two forms of spectral selectivity. Our results expand the roles of HCN channels to theta-frequency selectivity in the spike initiation dynamics, apart from underscoring the critical role of interactions among different ion channels in regulating neuronal physiology.NEW & NOTEWORTHY We had previously predicted, using computational analyses, that the spike-triggered average (STA) of hippocampal neurons would exhibit theta-frequency (4-10 Hz) spectral selectivity and would manifest coincidence detection capabilities for inputs in the gamma-frequency band (25-150 Hz). Here, we confirmed these predictions through direct electrophysiological recordings of STA from rat CA1 pyramidal neurons and demonstrate that blocking HCN channels reduces the frequency of STA spectral selectivity to the delta-frequency range (0.5-4 Hz).

Investigating the impact of electrical stimulation temporal distribution on cortical network responses

Background

The brain is continuously targeted by a wealth of stimuli with complex spatio-temporal patterns and has presumably evolved in order to cope with those inputs in an optimal way. Previous studies investigating the response capabilities of either single neurons or intact sensory systems to external stimulation demonstrated that stimuli temporal distribution is an important, if often overlooked, parameter.

Results

In this study we investigated how cortical networks plated over micro-electrode arrays respond to different stimulation sequences in which inter-pulse intervals followed a 1/f^β distribution, for different values of β ranging from 0 to ∞. Cross-correlation analysis revealed that network activity preferentially synchronizes with external input sequences featuring β closer to 1 and, in any case, never for regular (i.e. fixed-frequency) stimulation sequences. We then tested the interplay between different average stimulation frequencies (based on the intrinsic firing/bursting frequency of the network) for two selected values of β, i.e. 1 (scale free) and ∞ (regular). In general, we observed no preference for stimulation frequencies matching the endogenous rhythms of the network. Moreover, we found that in case of regular stimulation the capability of the network to follow the stimulation sequence was negatively correlated to the absolute stimulation frequency, whereas using scale-free stimulation cross-correlation between input and output sequences was independent from average input frequency.

Conclusions

Our results point out that the preference for a scale-free distribution of the stimuli is observed also at network level and should be taken into account in designing more efficient protocols for neuromodulation purposes.

Electronic supplementary material

The online version of this article (doi:10.1186/s12868-017-0366-z) contains supplementary material, which is available to authorized users.

Predicting the response of striatal spiny neurons to sinusoidal input

Spike-timing effects of small-amplitude sinusoidal currents were measured in mouse striatal spiny neurons firing repetitively. Spike-timing reliability varied with the stimulus frequency. For frequencies near the cell's firing rate, the cells altered firing rate to match the stimulus and became phase locked to it. The stimulus phase of firing during lock depended on the stimulus frequency relative to the cell's unperturbed firing rate. Interspike intervals during sinusoidal stimulation were predicted using an iterative map constructed from the cells' phase-resetting curve. Variability of interspike intervals was reduced by stimulation at all frequencies higher than about half the cell's unperturbed rate, and interspike intervals were accurately predicted by the map. Long sequences of spike times were predicted by iterating on the map. The accuracy of that prediction varied with frequency. Spike time predictability was highest near and during phase lock. The map predicted the phase of firing on the input and its dependence on stimulus frequency. Prediction errors, when they occurred, were of two kinds: unpredicted variation in interspike interval from intrinsic cell noise and accumulation of prediction errors from previous interspike intervals. Each type of prediction error arose from a different mechanism, and their impact was also predicted from the phase model. When two oscillatory input currents were presented simultaneously, striatal neurons responded selectively to only one of them, the one closest in frequency to the cell's unperturbed firing rate. Their spike times encoded the frequency and phase of that single oscillatory input.NEW & NOTEWORTHY During repetitive firing, the timing of action potentials is determined by the interaction between the input and voltage-sensitive currents throughout the interspike interval. This interaction is encapsulated in the neuron's phase-resetting curve. The phase-resetting curve predicted spike timing to small sinusoidal currents over a wide range of stimulus frequencies. Firing patterns were most sensitive to oscillatory components near the cell's own firing rate, even in the presence of noise and other inputs.

Strings on a Violin: Location Dependence of Frequency Tuning in Active Dendrites

Strings on a violin are tuned to generate distinct sound frequencies in a manner that is firmly dependent on finger location along the fingerboard. Sound frequencies emerging from different violins could be very different based on their architecture, the nature of strings and their tuning. Analogously, active neuronal dendrites, dendrites endowed with active channel conductances, are tuned to distinct input frequencies in a manner that is dependent on the dendritic location of the synaptic inputs. Further, disparate channel expression profiles and differences in morphological characteristics could result in dendrites on different neurons of the same subtype tuned to distinct frequency ranges. Alternately, similar location-dependence along dendritic structures could be achieved through disparate combinations of channel profiles and morphological characteristics, leading to degeneracy in active dendritic spectral tuning. Akin to strings on a violin being tuned to different frequencies than those on a viola or a cello, different neuronal subtypes exhibit distinct channel profiles and disparate morphological characteristics endowing each neuronal subtype with unique location-dependent frequency selectivity. Finally, similar to the tunability of musical instruments to elicit distinct location-dependent sounds, neuronal frequency selectivity and its location-dependence are tunable through activity-dependent plasticity of ion channels and morphology. In this morceau, we explore the origins of neuronal frequency selectivity, and survey the literature on the mechanisms behind the emergence of location-dependence in distinct forms of frequency tuning. As a coda to this composition, we present some future directions for this exciting convergence of biophysical mechanisms that endow a neuron with frequency multiplexing capabilities.

Systems-based analysis of dendritic nonlinearities reveals temporal feature extraction in mouse L5 cortical neurons

What do dendritic nonlinearities tell a neuron about signals injected into the dendrite? Linear and nonlinear dendritic components affect how time-varying inputs are transformed into action potentials (APs), but the relative contribution of each component is unclear. We developed a novel systems-identification approach to isolate the nonlinear response of layer 5 pyramidal neuron dendrites in mouse prefrontal cortex in response to dendritic current injections. We then quantified the nonlinear component and its effect on the soma, using functional models composed of linear filters and static nonlinearities. Both noise and waveform current injections revealed linear and nonlinear components in the dendritic response. The nonlinear component consisted of fast Na⁺ spikes that varied in amplitude 10-fold in a single neuron. A functional model reproduced the timing and amplitude of the dendritic spikes and revealed that they were selective to a preferred input dynamic (~4.5 ms rise time). The selectivity of the dendritic spikes became wider in the presence of additive noise, which was also predicted by the functional model. A second functional model revealed that the dendritic spikes were weakly boosted before being linearly integrated at the soma. For both our noise and waveform dendritic input, somatic APs were dependent on the somatic integration of the stimulus, followed a subset of large dendritic spikes, and were selective to the same input dynamics preferred by the dendrites. Our results suggest that the amplitude of fast dendritic spikes conveys information about high-frequency features in the dendritic input, which is then combined with low-frequency somatic integration.NEW & NOTEWORTHY The nonlinear response of layer 5 mouse pyramidal dendrites was isolated with a novel systems-based approach. In response to dendritic current injections, the nonlinear component contained mostly fast, variable-amplitude, Na⁺ spikes. A functional model accounted for the timing and amplitude of the dendritic spikes and revealed that dendritic spikes are selective to a preferred input dynamic, which was verified experimentally. Thus, fast dendritic nonlinearities behave as high-frequency feature detectors that influence somatic action potentials.

Encoding in Balanced Networks: Revisiting Spike Patterns and Chaos in Stimulus-Driven Systems

Highly connected recurrent neural networks often produce chaotic dynamics, meaning their precise activity is sensitive to small perturbations. What are the consequences of chaos for how such networks encode streams of temporal stimuli? On the one hand, chaos is a strong source of randomness, suggesting that small changes in stimuli will be obscured by intrinsically generated variability. On the other hand, recent work shows that the type of chaos that occurs in spiking networks can have a surprisingly low-dimensional structure, suggesting that there may be room for fine stimulus features to be precisely resolved. Here we show that strongly chaotic networks produce patterned spikes that reliably encode time-dependent stimuli: using a decoder sensitive to spike times on timescales of 10’s of ms, one can easily distinguish responses to very similar inputs. Moreover, recurrence serves to distribute signals throughout chaotic networks so that small groups of cells can encode substantial information about signals arriving elsewhere. A conclusion is that the presence of strong chaos in recurrent networks need not exclude precise encoding of temporal stimuli via spike patterns.

Recurrently connected populations of excitatory and inhibitory neurons found in cortex are known to produce rich and irregular spiking activity, with complex trial-to-trial variability in response to input stimuli. Many theoretical studies found this firing regime to be associated with chaos, where tiny perturbations explode to impact subsequent neural activity. As a result, the precise spiking patterns produced by such networks would be expected to be too fragile to carry any valuable information about stimuli, since inevitable sources of noise such as synaptic failure or ion channel fluctuations would be amplified by chaotic dynamics on repeated trials. In this article we revisit the implications of chaos in input-driven networks and directly measure its impact on evoked population spike patterns. We find that chaotic network dynamics can, in fact, produce highly patterned spiking activity which can be used by a simple decoder to perform input-classification tasks. This can be explained by the presence of low-dimensional, input-specific chaotic attractors, leading to a form of trial-to-trial variability that is intermittent, rather than uniformly random. We propose that chaos is a manageable by-product of recurrent connectivity, which serves to efficiently distribute information about stimuli throughout a network.

Stochastic Network Models in Neuroscience: A Festschrift for Jack Cowan. Introduction to the Special Issue

Jack Cowan's remarkable career has spanned, and molded, the development of neuroscience as a quantitative and mathematical discipline combining deep theoretical contributions, rigorous mathematical work and groundbreaking biological insights. The Banff International Research Station hosted a workshop in his honor, on Stochastic Network Models of Neocortex, July 17-24, 2014. This accompanying Festschrift celebrates Cowan's contributions by assembling current research in stochastic phenomena in neural networks. It combines historical perspectives with new results including applications to epilepsy, path-integral methods, stochastic synchronization, higher-order correlation analysis, and pattern formation in visual cortex.

How Do Efficient Coding Strategies Depend on Origins of Noise in Neural Circuits?

Neural circuits reliably encode and transmit signals despite the presence of noise at multiple stages of processing. The efficient coding hypothesis, a guiding principle in computational neuroscience, suggests that a neuron or population of neurons allocates its limited range of responses as efficiently as possible to best encode inputs while mitigating the effects of noise. Previous work on this question relies on specific assumptions about where noise enters a circuit, limiting the generality of the resulting conclusions. Here we systematically investigate how noise introduced at different stages of neural processing impacts optimal coding strategies. Using simulations and a flexible analytical approach, we show how these strategies depend on the strength of each noise source, revealing under what conditions the different noise sources have competing or complementary effects. We draw two primary conclusions: (1) differences in encoding strategies between sensory systems-or even adaptational changes in encoding properties within a given system-may be produced by changes in the structure or location of neural noise, and (2) characterization of both circuit nonlinearities as well as noise are necessary to evaluate whether a circuit is performing efficiently.

Perineuronal Nets Enhance the Excitability of Fast-Spiking Neurons

Perineuronal nets (PNNs) are specialized complexes of extracellular matrix molecules that surround the somata of fast-spiking neurons throughout the vertebrate brain. PNNs are particularly prevalent throughout the auditory brainstem, which transmits signals with high speed and precision. It is unknown whether PNNs contribute to the fast-spiking ability of the neurons they surround. Whole-cell recordings were made from medial nucleus of the trapezoid body (MNTB) principal neurons in acute brain slices from postnatal day 21 (P21) to P27 mice. PNNs were degraded by incubating slices in chondroitinase ABC (ChABC) and were compared to slices that were treated with a control enzyme (penicillinase). ChABC treatment did not affect the ability of MNTB neurons to fire at up to 1000 Hz when driven by current pulses. However, f–I (frequency–intensity) curves constructed by injecting Gaussian white noise currents superimposed on DC current steps showed that ChABC treatment reduced the gain of spike output. An increase in spike threshold may have contributed to this effect, which is consistent with the observation that spikes in ChABC-treated cells were delayed relative to control-treated cells. In addition, parvalbumin-expressing fast-spiking cortical neurons in >P70 slices that were treated with ChABC also had reduced excitability and gain. The development of PNNs around somata of fast-spiking neurons may be essential for fast and precise sensory transmission and synaptic inhibition in the brain.

Impact of stimuli distribution on neural network responses

In this study we investigated the relationship between stimulus regularity and observed responses in a biological neural network of dissociated cortical rat neurons plated over Micro-Electrode Arrays (MEAs). In particular, the intervals between identical stimuli in our experiments followed a 1/f(β) distribution, and regularity increased with the value of β (values tested were 0, 0.5, 1, 1.5, ∞). Comparisons occurred on the correlation between low-passed (rectangular window, 0.1s in length) stimulation trains and network-wide spike trains. Our results show that cultures are largely unable to synchronize network-wide responses with regular stimulation at the considered stimulation rate (0.5Hz), while this occurs to a much higher degree for irregular stimulations.

Philosophy of the Spike: Rate-Based vs. Spike-Based Theories of the Brain

Does the brain use a firing rate code or a spike timing code? Considering this controversial question from an epistemological perspective, I argue that progress has been hampered by its problematic phrasing. It takes the perspective of an external observer looking at whether those two observables vary with stimuli, and thereby misses the relevant question: which one has a causal role in neural activity? When rephrased in a more meaningful way, the rate-based view appears as an ad hoc methodological postulate, one that is practical but with virtually no empirical or theoretical support.

Postnatal development attunes olfactory bulb mitral cells to high-frequency signaling

Mitral cells (MCs) are a major class of principal neurons in the vertebrate olfactory bulb, conveying odor-evoked activity from the peripheral sensory neurons to olfactory cortex. Previous work has described the development of MC morphology and connectivity during the first few weeks of postnatal development. However, little is known about the postnatal development of MC intrinsic biophysical properties. To understand stimulus encoding in the developing olfactory bulb, we have therefore examined the development of MC intrinsic biophysical properties in acute slices from postnatal day (P)7-P35 mice. Across development, we observed systematic changes in passive membrane properties and action potential waveforms consistent with a developmental increase in sodium and potassium conductances. We further observed developmental decreases in hyperpolarization-evoked membrane potential sag and firing regularity, extending recent links between MC sag heterogeneity and firing patterns. We then applied a novel combination of statistical analyses to examine how the evolution of these intrinsic biophysical properties specifically influenced the representation of fluctuating stimuli by MCs. We found that immature MCs responded to frozen fluctuating stimuli with lower firing rates, lower spike-time reliability, and lower between-cell spike-time correlations than more mature MCs. Analysis of spike-triggered averages revealed that these changes in spike timing were driven by a developmental shift from broad integration of inputs to more selective detection of coincident inputs. Consistent with this shift, generalized linear model fits to MC firing responses demonstrated an enhanced encoding of high-frequency stimulus features by mature MCs.

Spike-Triggered Covariance Analysis Reveals Phenomenological Diversity of Contrast Adaptation in the Retina

When visual contrast changes, retinal ganglion cells adapt by adjusting their sensitivity as well as their temporal filtering characteristics. The latter has classically been described by contrast-induced gain changes that depend on temporal frequency. Here, we explored a new perspective on contrast-induced changes in temporal filtering by using spike-triggered covariance analysis to extract multiple parallel temporal filters for individual ganglion cells. Based on multielectrode-array recordings from ganglion cells in the isolated salamander retina, we found that contrast adaptation of temporal filtering can largely be captured by contrast-invariant sets of filters with contrast-dependent weights. Moreover, differences among the ganglion cells in the filter sets and their contrast-dependent contributions allowed us to phenomenologically distinguish three types of filter changes. The first type is characterized by newly emerging features at higher contrast, which can be reproduced by computational models that contain response-triggered gain-control mechanisms. The second type follows from stronger adaptation in the Off pathway as compared to the On pathway in On-Off-type ganglion cells. Finally, we found that, in a subset of neurons, contrast-induced filter changes are governed by particularly strong spike-timing dynamics, in particular by pronounced stimulus-dependent latency shifts that can be observed in these cells. Together, our results show that the contrast dependence of temporal filtering in retinal ganglion cells has a multifaceted phenomenology and that a multi-filter analysis can provide a useful basis for capturing the underlying signal-processing dynamics.

Our sensory systems have to process stimuli under a wide range of environmental conditions. To cope with this challenge, the involved neurons adapt by adjusting their signal processing to the recently encountered intensity range. In the visual system, one finds, for example, that higher visual contrast leads to changes in how visual signals are temporally filtered, making signal processing faster and more band-pass-like at higher contrast. By analyzing signals from neurons in the retina of salamanders, we here found that these adaptation effects can be described by a fixed set of filters, independent of contrast, whose relative contributions change with contrast. Also, we found that different phenomena contribute to this adaptation. In particular, some cells change their relative sensitivity to light increments and light decrements, whereas other cells are influenced by a strong contrast-dependence of the exact timing of their responses. Our results show that contrast adaptation in the retina is not an entirely homogeneous phenomenon, and that models with multiple filters can help in characterizing sensory adaptation.

Irregular spiking of pyramidal neurons organizes as scale-invariant neuronal avalanches in the awake state

Spontaneous fluctuations in neuronal activity emerge at many spatial and temporal scales in cortex. Population measures found these fluctuations to organize as scale-invariant neuronal avalanches, suggesting cortical dynamics to be critical. Macroscopic dynamics, though, depend on physiological states and are ambiguous as to their cellular composition, spatiotemporal origin, and contributions from synaptic input or action potential (AP) output. Here, we study spontaneous firing in pyramidal neurons (PNs) from rat superficial cortical layers in vivo and in vitro using 2-photon imaging. As the animal transitions from the anesthetized to awake state, spontaneous single neuron firing increases in irregularity and assembles into scale-invariant avalanches at the group level. In vitro spike avalanches emerged naturally yet required balanced excitation and inhibition. This demonstrates that neuronal avalanches are linked to the global physiological state of wakefulness and that cortical resting activity organizes as avalanches from firing of local PN groups to global population activity.

DOI:http://dx.doi.org/10.7554/eLife.07224.001

Even when we are not engaged in any specific task, the brain shows coordinated patterns of spontaneous activity that can be monitored using electrodes placed on the scalp. This resting activity shapes the way that the brain responds to subsequent stimuli. Changes in resting activity patterns are seen in various neurological and psychiatric disorders, as well as in healthy individuals following sleep deprivation.

The brain's outer layer is known as the cortex. On a large scale, when monitoring many thousands of neurons, resting activity in the cortex demonstrates propagation in the brain in an organized manner. Specifically, resting activity was found to organize as so-called neuronal avalanches, in which large bursts of neuronal activity are grouped with medium-sized and smaller bursts in a very characteristic order. In fact, the sizes of these bursts—that is, the number of neurons that fire—are found to be scale-invariant, that is, the ratio of large bursts to medium-sized bursts is the same as that of medium-sized to small bursts. Such scale-invariance suggests that neuronal bursts are not independent of one another. However, it is largely unclear how neuronal avalanches arise from individual neurons, which fire simply in a noisy, irregular manner.

Bellay, Klaus et al. have now provided insights into this process by examining patterns of firing of a particular type of neuron—known as a pyramidal cell—in the cortex of rats as they recover from anesthesia. As the animals awaken, the firing of individual pyramidal cells in the cortex becomes even more irregular than under anesthesia. However, by considering the activity of a group of these neurons, Bellay, Klaus et al. realized that it is this more irregular firing that gives rise to neuronal avalanches, and that this occurs only when the animals are awake. Further experiments on individual pyramidal cells grown in the laboratory confirmed that neuronal avalanches emerge spontaneously from the irregular firing of individual neurons. These avalanches depend on there being a balance between two types of activity among the cells: ‘excitatory’ activity that causes other neurons to fire, and ‘inhibitory’ activity that prevents neuronal firing.

Given that resting activity influences the brain's responses to the outside world, the origins of neuronal avalanches are likely to provide clues about the way the brain processes information. Future experiments should also examine the possibility that the emergence of neuronal avalanches marks the transition from unconsciousness to wakefulness within the brain.

DOI:http://dx.doi.org/10.7554/eLife.07224.002

Optimal stimulus waveforms for eliciting a spike: How close is the spike-triggered average?

Computing the average input stimulus preceding a spike, the spike-triggered average (STA), has been a powerful tool for discovering a neuron's 'preferred' stimulus feature that enables efficient encoding of sensory information. Recent work in the squid giant axon has shown that STA waveforms can be remarkably similar to the energetically optimal stimulus waveforms for eliciting a spike. In the present study, we show using the Hodgkin-Huxley model that the STA can deviate from the global optimal solution if there is averaging of multiple solutions across different time scales and of multiple modes of spike induction. These findings inform attempts to develop model-free stochastic algorithms for finding energy-optimal stimuli, which is relevant to the efficient delivery of exogenous therapeutic stimuli in neurological diseases.

Microsecond-scale timing precision in rodent trigeminal primary afferents

Communication in the nervous system occurs by spikes: the timing precision with which spikes are fired is a fundamental limit on neural information processing. In sensory systems, spike-timing precision is constrained by first-order neurons. We found that spike-timing precision of trigeminal primary afferents in rats and mice is limited both by stimulus speed and by electrophysiological sampling rate. High-speed video of behaving mice revealed whisker velocities of at least 17,000°/s, so we delivered an ultrafast "ping" (>50,000°/s) to single whiskers and sampled primary afferent activity at 500 kHz. Median spike jitter was 17.4 μs; 29% of neurons had spike jitter < 10 μs. These results indicate that the input stage of the trigeminal pathway has extraordinary spike-timing precision and very high potential information capacity. This timing precision ranks among the highest in biology.

An Empirical Model for Reliable Spiking Activity

Understanding a neuron's transfer function, which relates a neuron's inputs to its outputs, is essential for understanding the computational role of single neurons. Recently, statistical models, based on point processes and using generalized linear model (GLM) technology, have been widely applied to predict dynamic neuronal transfer functions. However, the standard version of these models fails to capture important features of neural activity, such as responses to stimuli that elicit highly reliable trial-to-trial spiking. Here, we consider a generalization of the usual GLM that incorporates nonlinearity by modeling reliable and nonreliable spikes as being generated by distinct stimulus features. We develop and apply these models to spike trains from olfactory bulb mitral cells recorded in vitro. We find that spike generation in these neurons is better modeled when reliable and unreliable spikes are considered separately and that this effect is most pronounced for neurons with a large number of both reliable and unreliable spikes.

For things needing your attention: the role of neocortical gamma in sensory perception

Two general classes of hypotheses for the role for gamma oscillations in sensation are those that predict gamma facilitates signal amplification through local synchronization of a distinct ensemble, and those that predict gamma modulates fine temporal relationships between neurons to represent information. Correlative evidence has been offered for and against these hypotheses. A recent study in which gamma was optogenetically entrained by driving fast-spiking interneurons showed enhanced sensory detection of harder-to-perceive stimuli, those that benefit most from attention, in agreement with the amplification hypotheses. These findings are supported by similar studies employing less specific optogenetic patterns or single neuron stimulation, but contrast with findings based on direct optogenetic stimulation of pyramidal neurons. Key next steps for this topic are described.

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.

Principal dynamic mode analysis of the Hodgkin-Huxley equations

We develop an autoregressive model framework based on the concept of Principal Dynamic Modes (PDMs) for the process of action potential (AP) generation in the excitable neuronal membrane described by the Hodgkin-Huxley (H-H) equations. The model's exogenous input is injected current, and whenever the membrane potential output exceeds a specified threshold, it is fed back as a second input. The PDMs are estimated from the previously developed Nonlinear Autoregressive Volterra (NARV) model, and represent an efficient functional basis for Volterra kernel expansion. The PDM-based model admits a modular representation, consisting of the forward and feedback PDM bases as linear filterbanks for the exogenous and autoregressive inputs, respectively, whose outputs are then fed to a static nonlinearity composed of polynomials operating on the PDM outputs and cross-terms of pair-products of PDM outputs. A two-step procedure for model reduction is performed: first, influential subsets of the forward and feedback PDM bases are identified and selected as the reduced PDM bases. Second, the terms of the static nonlinearity are pruned. The first step reduces model complexity from a total of 65 coefficients to 27, while the second further reduces the model coefficients to only eight. It is demonstrated that the performance cost of model reduction in terms of out-of-sample prediction accuracy is minimal. Unlike the full model, the eight coefficient pruned model can be easily visualized to reveal the essential system components, and thus the data-derived PDM model can yield insight into the underlying system structure and function.

Systematic analysis of the contributions of stochastic voltage gated channels to neuronal noise

Electrical signaling in neurons is mediated by the opening and closing of large numbers of individual ion channels. The ion channels' state transitions are stochastic and introduce fluctuations in the macroscopic current through ion channel populations. This creates an unavoidable source of intrinsic electrical noise for the neuron, leading to fluctuations in the membrane potential and spontaneous spikes. While this effect is well known, the impact of channel noise on single neuron dynamics remains poorly understood. Most results are based on numerical simulations. There is no agreement, even in theoretical studies, on which ion channel type is the dominant noise source, nor how inclusion of additional ion channel types affects voltage noise. Here we describe a framework to calculate voltage noise directly from an arbitrary set of ion channel models, and discuss how this can be use to estimate spontaneous spike rates.

Context-dependent coding in single neurons

The linear-nonlinear cascade model (LN model) has proven very useful in representing a neural system's encoding properties, but has proven less successful in reproducing the firing patterns of individual neurons whose behavior is strongly dependent on prior firing history. While the cell's behavior can still usefully be considered as feature detection acting on a fluctuating input, some of the coding capacity of the cell is taken up by the increased firing rate due to a constant "driving" direct current (DC) stimulus. Furthermore, both the DC input and the post-spike refractory period generate regular firing, reducing the spike-timing entropy available for encoding time-varying fluctuations. In this paper, we address these issues, focusing on the example of motoneurons in which an afterhyperpolarization (AHP) current plays a dominant role regularizing firing behavior. We explore the accuracy and generalizability of several alternative models for single neurons under changes in DC and variance of the stimulus input. We use a motoneuron simulation to compare coding models in neurons with and without the AHP current. Finally, we quantify the tradeoff between instantaneously encoding information about fluctuations and about the DC.

Disrupting information coding via block of 4-AP-sensitive potassium channels

Recent interest has emerged on the role of intrinsic biophysical diversity in neuronal coding. An important question in neurophysiology is understanding which voltage-gated ion channels are responsible for this diversity and how variable expression or activity of one class of ion channels across neurons of a single type affects they way populations carry information. In mitral cells in the olfactory bulb of mice, we found that biophysical diversity was conferred in part by 4-aminopyridine (4-AP)-sensitive potassium channels and reduced following block of those channels. When populations of mitral cells were stimulated with identical inputs, the diversity exhibited in their output spike patterns reduced with the addition of 4-AP, decreasing the stimulus information carried by ensembles of 15 neurons from 437 ± 15 to 397 ± 19 bits/s. Decreases in information were due to reduction in the diversity of population spike patterns generated in response to different features of the stimulus, suggesting that the coding capacity of a population can be altered by changes in the function of single ion channel types.

Neuronal spike timing adaptation described with a fractional leaky integrate-and-fire model

The voltage trace of neuronal activities can follow multiple timescale dynamics that arise from correlated membrane conductances. Such processes can result in power-law behavior in which the membrane voltage cannot be characterized with a single time constant. The emergent effect of these membrane correlations is a non-Markovian process that can be modeled with a fractional derivative. A fractional derivative is a non-local process in which the value of the variable is determined by integrating a temporal weighted voltage trace, also called the memory trace. Here we developed and analyzed a fractional leaky integrate-and-fire model in which the exponent of the fractional derivative can vary from 0 to 1, with 1 representing the normal derivative. As the exponent of the fractional derivative decreases, the weights of the voltage trace increase. Thus, the value of the voltage is increasingly correlated with the trajectory of the voltage in the past. By varying only the fractional exponent, our model can reproduce upward and downward spike adaptations found experimentally in neocortical pyramidal cells and tectal neurons in vitro. The model also produces spikes with longer first-spike latency and high inter-spike variability with power-law distribution. We further analyze spike adaptation and the responses to noisy and oscillatory input. The fractional model generates reliable spike patterns in response to noisy input. Overall, the spiking activity of the fractional leaky integrate-and-fire model deviates from the spiking activity of the Markovian model and reflects the temporal accumulated intrinsic membrane dynamics that affect the response of the neuron to external stimulation.

Spike adaptation is a property of most neurons. When spike time adaptation occurs over multiple time scales, the dynamics can be described by a power-law. We study the computational properties of a leaky integrate-and-fire model with power-law adaptation. Instead of explicitly modeling the adaptation process by the contribution of slowly changing conductances, we use a fractional temporal derivative framework. The exponent of the fractional derivative represents the degree of adaptation of the membrane voltage, where 1 is the normal leaky integrator while values less than 1 produce increasing correlations in the voltage trace. The temporal correlation is interpreted as a memory trace that depends on the value of the fractional derivative. We identify the memory trace in the fractional model as the sum of the instantaneous differentiation weighted by a function that depends on the fractional exponent, and it provides non-local information to the incoming stimulus. The spiking dynamics of the fractional leaky integrate-and-fire model show memory dependence that can result in downward or upward spike adaptation. Our model provides a framework for understanding how long-range membrane voltage correlations affect spiking dynamics and information integration in neurons.