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

Pulse-frequency-dependent resonance in a population of pyramidal neuron models

Stochastic resonance is known as a phenomenon whereby information transmission of weak signal or subthreshold stimuli can be enhanced by additive random noise with a suitable intensity. Another phenomenon induced by applying deterministic pulsatile electric stimuli with a pulse frequency, commonly used for deep brain stimulation (DBS), was also shown to improve signal-to-noise ratio in neuron models. The objective of this study was to test the hypothesis that pulsatile high-frequency stimulation could improve the detection of both sub- and suprathreshold synaptic stimuli by tuning the frequency of the stimulation in a population of pyramidal neuron models. Computer simulations showed that mutual information estimated from a population of neural spike trains displayed a typical resonance curve with a peak value of the pulse frequency at 80-120 Hz, similar to those utilized for DBS in clinical situations. It is concluded that a "pulse-frequency-dependent resonance" (PFDR) can enhance information transmission over a broad range of synaptically connected networks. Since the resonance frequency matches that used clinically, PFDR could contribute to the mechanism of the therapeutic effect of DBS.

Prefrontal oscillations modulate the propagation of neuronal activity required for working memory

Cognition involves using attended information, maintained in working memory (WM), to guide action. During a cognitive task, a correct response requires flexible, selective gating so that only the appropriate information flows from WM to downstream effectors that carry out the response. In this work, we used biophysically-detailed modeling to explore the hypothesis that network oscillations in prefrontal cortex (PFC), leveraging local inhibition, can independently gate responses to items in WM. The key role of local inhibition was to control the period between spike bursts in the outputs, and to produce an oscillatory response no matter whether the WM item was maintained in an asynchronous or oscillatory state. We found that the WM item that induced an oscillatory population response in the PFC output layer with the shortest period between spike bursts was most reliably propagated. The network resonant frequency (i.e., the input frequency that produces the largest response) of the output layer can be flexibly tuned by varying the excitability of deep layer principal cells. Our model suggests that experimentally-observed modulation of PFC beta-frequency (15-30 Hz) and gamma-frequency (30-80 Hz) oscillations could leverage network resonance and local inhibition to govern the flexible routing of signals in service to cognitive processes like gating outputs from working memory and the selection of rule-based actions. Importantly, we show for the first time that nonspecific changes in deep layer excitability can tune the output gate's resonant frequency, enabling the specific selection of signals encoded by populations in asynchronous or fast oscillatory states. More generally, this represents a dynamic mechanism by which adjusting network excitability can govern the propagation of asynchronous and oscillatory signals throughout neocortex.

The Impact of Frequency Scale on the Response Sensitivity and Reliability of Cortical Neurons to 1/fβ Input Signals

What type of principle features intrinsic inside of the fluctuated input signals could drive neurons with the maximal excitations is one of the crucial neural coding issues. In this article, we examined both experimentally and theoretically the cortical neuronal responsivity (including firing rate and spike timing reliability) to input signals with different intrinsic correlational statistics (e.g., white-type noise, showed 1/f0 power spectrum, pink noise 1/f, and brown noises 1/f2) and different frequency ranges. Our results revealed that the response sensitivity and reliability of cortical neurons is much higher in response to 1/f noise stimuli with long-term correlations than 1/f0 with short-term correlations for a broad frequency range, and also higher than 1/f2 for all frequency ranges. In addition, we found that neuronal sensitivity diverges to opposite directions for 1/f noise comparing with 1/f0 white noise as a function of cutoff frequency of input signal. As the cutoff frequency is progressively increased from 50 to 1,000 Hz, the neuronal responsiveness increased gradually for 1/f noise, while decreased exponentially for white noise. Computational simulations of a general cortical model revealed that, neuronal sensitivity and reliability to input signal statistics was majorly dominated by fast sodium inactivation, potassium activation, and membrane time constants.

How single neuron properties shape chaotic dynamics and signal transmission in random neural networks

While most models of randomly connected neural networks assume single-neuron models with simple dynamics, neurons in the brain exhibit complex intrinsic dynamics over multiple timescales. We analyze how the dynamical properties of single neurons and recurrent connections interact to shape the effective dynamics in large randomly connected networks. A novel dynamical mean-field theory for strongly connected networks of multi-dimensional rate neurons shows that the power spectrum of the network activity in the chaotic phase emerges from a nonlinear sharpening of the frequency response function of single neurons. For the case of two-dimensional rate neurons with strong adaptation, we find that the network exhibits a state of "resonant chaos", characterized by robust, narrow-band stochastic oscillations. The coherence of stochastic oscillations is maximal at the onset of chaos and their correlation time scales with the adaptation timescale of single units. Surprisingly, the resonance frequency can be predicted from the properties of isolated neurons, even in the presence of heterogeneity in the adaptation parameters. In the presence of these internally-generated chaotic fluctuations, the transmission of weak, low-frequency signals is strongly enhanced by adaptation, whereas signal transmission is not influenced by adaptation in the non-chaotic regime. Our theoretical framework can be applied to other mechanisms at the level of single neurons, such as synaptic filtering, refractoriness or spike synchronization. These results advance our understanding of the interaction between the dynamics of single units and recurrent connectivity, which is a fundamental step toward the description of biologically realistic neural networks.

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

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

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.

Accuracy and Efficiency in Fixed-Point Neural ODE Solvers

Simulation of neural behavior on digital architectures often requires the solution of ordinary differential equations (ODEs) at each step of the simulation. For some neural models, this is a significant computational burden, so efficiency is important. Accuracy is also relevant because solutions can be sensitive to model parameterization and time step. These issues are emphasized on fixed-point processors like the ARM unit used in the SpiNNaker architecture. Using the Izhikevich neural model as an example, we explore some solution methods, showing how specific techniques can be used to find balanced solutions. We have investigated a number of important and related issues, such as introducing explicit solver reduction (ESR) for merging an explicit ODE solver and autonomous ODE into one algebraic formula, with benefits for both accuracy and speed; a simple, efficient mechanism for cancelling the cumulative lag in state variables caused by threshold crossing between time steps; an exact result for the membrane potential of the Izhikevich model with the other state variable held fixed. Parametric variations of the Izhikevich neuron show both similarities and differences in terms of algorithms and arithmetic types that perform well, making an overall best solution challenging to identify, but we show that particular cases can be improved significantly using the techniques described. Using a 1 ms simulation time step and 32-bit fixed-point arithmetic to promote real-time performance, one of the second-order Runge-Kutta methods looks to be the best compromise; Midpoint for speed or Trapezoid for accuracy. SpiNNaker offers an unusual combination of low energy use and real-time performance, so some compromises on accuracy might be expected. However, with a careful choice of approach, results comparable to those of general-purpose systems should be possible in many realistic cases.

Oscillations can reconcile slowly changing stimuli with short neuronal integration and STDP timescales

Oscillatory brain activity has been widely reported experimentally, yet its functional roles, if any, are still under debate. In this review we argue two things: firstly, thanks to oscillations, even slowly changing stimuli can be encoded in precise relative spike times, decodable by downstream "coincidence detector" neurons in a feedforward manner. Secondly, the required connectivity to do so can spontaneously emerge with spike timing-dependent plasticity (STDP), in an unsupervised manner. The key here is that a common oscillatory drive enables neurons to remain under a fluctuation-driven regime. In this regime spike time jitter does not accumulate and can thus be lower than the intrinsic timescales of stimulus fluctuations, which leads to so-called "temporal encoding". Furthermore, the oscillatory drive formats the spikes in discrete oversampling volleys, and the relative spike times between neurons indicate the eventual differences in their activation levels. The oversampling accelerates the STDP-based learning for downstream neurons. After learning, readout only takes one oscillatory cycle. Finally, we also discuss experimental evidence, and the question of how the theory is complementary to the so-called "communication through coherence" theory.

A computational study of spike time reliability in two types of threshold dynamics

Spike time reliability (STR) refers to the phenomenon in which repetitive applications of a frozen copy of one stochastic signal to a neuron trigger spikes with reliable timing while a constant signal fails to do so. Observed and explored in numerous experimental and theoretical studies, STR is a complex dynamic phenomenon depending on the nature of external inputs as well as intrinsic properties of a neuron. The neuron under consideration could be either quiescent or spontaneously spiking in the absence of the external stimulus. Focusing on the situation in which the unstimulated neuron is quiescent but close to a switching point to oscillations, we numerically analyze STR treating each spike occurrence as a time localized event in a model neuron. We study both the averaged properties as well as individual features of spike-evoking epochs (SEEs). The effects of interactions between spikes is minimized by selecting signals that generate spikes with relatively long interspike intervals (ISIs). Under these conditions, the frequency content of the input signal has little impact on STR. We study two distinct cases, Type I in which the f-I relation (f for frequency, I for applied current) is continuous and Type II where the f-I relation exhibits a jump. STR in the two types shows a number of similar features and differ in some others. SEEs that are capable of triggering spikes show great variety in amplitude and time profile. On average, reliable spike timing is associated with an accelerated increase in the "action" of the signal as a threshold for spike generation is approached. Here, "action" is defined as the average amount of current delivered during a fixed time interval. When individual SEEs are studied, however, their time profiles are found important for triggering more precisely timed spikes. The SEEs that have a more favorable time profile are capable of triggering spikes with higher precision even at lower action levels.

A phase transition in the first passage of a Brownian process through a fluctuating boundary with implications for neural coding

Finding the first time a fluctuating quantity reaches a given boundary is a deceptively simple-looking problem of vast practical importance in physics, biology, chemistry, neuroscience, economics, and industrial engineering. Problems in which the bound to be traversed is itself a fluctuating function of time include widely studied problems in neural coding, such as neuronal integrators with irregular inputs and internal noise. We show that the probability p(t) that a Gauss-Markov process will first exceed the boundary at time t suffers a phase transition as a function of the roughness of the boundary, as measured by its Hölder exponent H. The critical value occurs when the roughness of the boundary equals the roughness of the process, so for diffusive processes the critical value is Hc = 1/2. For smoother boundaries, H > 1/2, the probability density is a continuous function of time. For rougher boundaries, H < 1/2, the probability is concentrated on a Cantor-like set of zero measure: the probability density becomes divergent, almost everywhere either zero or infinity. The critical point Hc = 1/2 corresponds to a widely studied case in the theory of neural coding, in which the external input integrated by a model neuron is a white-noise process, as in the case of uncorrelated but precisely balanced excitatory and inhibitory inputs. We argue that this transition corresponds to a sharp boundary between rate codes, in which the neural firing probability varies smoothly, and temporal codes, in which the neuron fires at sharply defined times regardless of the intensity of internal noise.

Computing with neural synchrony

Neurons communicate primarily with spikes, but most theories of neural computation are based on firing rates. Yet, many experimental observations suggest that the temporal coordination of spikes plays a role in sensory processing. Among potential spike-based codes, synchrony appears as a good candidate because neural firing and plasticity are sensitive to fine input correlations. However, it is unclear what role synchrony may play in neural computation, and what functional advantage it may provide. With a theoretical approach, I show that the computational interest of neural synchrony appears when neurons have heterogeneous properties. In this context, the relationship between stimuli and neural synchrony is captured by the concept of synchrony receptive field, the set of stimuli which induce synchronous responses in a group of neurons. In a heterogeneous neural population, it appears that synchrony patterns represent structure or sensory invariants in stimuli, which can then be detected by postsynaptic neurons. The required neural circuitry can spontaneously emerge with spike-timing-dependent plasticity. Using examples in different sensory modalities, I show that this allows simple neural circuits to extract relevant information from realistic sensory stimuli, for example to identify a fluctuating odor in the presence of distractors. This theory of synchrony-based computation shows that relative spike timing may indeed have computational relevance, and suggests new types of neural network models for sensory processing with appealing computational properties.

How does the brain compute? Traditional theories of neural computation describe the operating function of neurons in terms of average firing rates, with the timing of spikes bearing little information. However, numerous studies have shown that spike timing can convey information and that neurons are highly sensitive to synchrony in their inputs. Here I propose a simple spike-based computational framework, based on the idea that stimulus-induced synchrony can be used to extract sensory invariants (for example, the location of a sound source), which is a difficult task for classical neural networks. It relies on the simple remark that a series of repeated coincidences is in itself an invariant. Many aspects of perception rely on extracting invariant features, such as the spatial location of a time-varying sound, the identity of an odor with fluctuating intensity, the pitch of a musical note. I demonstrate that simple synchrony-based neuron models can extract these useful features, by using spiking models in several sensory modalities.

Spiking models for level-invariant encoding

Levels of ecological sounds vary over several orders of magnitude, but the firing rate and membrane potential of a neuron are much more limited in range. In binaural neurons of the barn owl, tuning to interaural delays is independent of level differences. Yet a monaural neuron with a fixed threshold should fire earlier in response to louder sounds, which would disrupt the tuning of these neurons. How could spike timing be independent of input level? Here I derive theoretical conditions for a spiking model to be insensitive to input level. The key property is a dynamic change in spike threshold. I then show how level invariance can be physiologically implemented, with specific ionic channel properties. It appears that these ingredients are indeed present in monaural neurons of the sound localization pathway of birds and mammals.

Stochastic resonance with a mixture of sub-and supra-threshold stimuli in a population of neuron models

This paper presents a novel type of stochastic resonance (SR) with a mixture of sub- and supra-threshold stimuli in a population of neuron models beyond regular SR and Supra-threshold SR (SSR) phenomena. We investigate through computer simulations if the novel type of SR can be observed or not, using the mutual information (MI) estimated from a population of neural spike trains as an index of information transmission. Computer simulations showed that the MI had a typical type of SR curves, even when the balance between sub-and supra-threshold stimuli was varied, suggesting the novel type of SR. Moreover, the peak of MI increased as the balance of supra-threshold stimuli got stronger, i.e., as the situation was getting close to the SSR from the regular SR. This finding could accelerate our understanding about how fluctuations play a role in processing information carried by a mixture of sub-and supra-threshold stimuli.

Spike-timing dependent plasticity and feed-forward input oscillations produce precise and invariant spike phase-locking

In the hippocampus and the neocortex, the coupling between local field potential (LFP) oscillations and the spiking of single neurons can be highly precise, across neuronal populations and cell types. Spike phase (i.e., the spike time with respect to a reference oscillation) is known to carry reliable information, both with phase-locking behavior and with more complex phase relationships, such as phase precession. How this precision is achieved by neuronal populations, whose membrane properties and total input may be quite heterogeneous, is nevertheless unknown. In this note, we investigate a simple mechanism for learning precise LFP-to-spike coupling in feed-forward networks – the reliable, periodic modulation of presynaptic firing rates during oscillations, coupled with spike-timing dependent plasticity. When oscillations are within the biological range (2–150 Hz), firing rates of the inputs change on a timescale highly relevant to spike-timing dependent plasticity (STDP). Through analytic and computational methods, we find points of stable phase-locking for a neuron with plastic input synapses. These points correspond to precise phase-locking behavior in the feed-forward network. The location of these points depends on the oscillation frequency of the inputs, the STDP time constants, and the balance of potentiation and de-potentiation in the STDP rule. For a given input oscillation, the balance of potentiation and de-potentiation in the STDP rule is the critical parameter that determines the phase at which an output neuron will learn to spike. These findings are robust to changes in intrinsic post-synaptic properties. Finally, we discuss implications of this mechanism for stable learning of spike-timing in the hippocampus.

Relative spike time coding and STDP-based orientation selectivity in the early visual system in natural continuous and saccadic vision: a computational model

We have built a phenomenological spiking model of the cat early visual system comprising the retina, the Lateral Geniculate Nucleus (LGN) and V1's layer 4, and established four main results (1) When exposed to videos that reproduce with high fidelity what a cat experiences under natural conditions, adjacent Retinal Ganglion Cells (RGCs) have spike-time correlations at a short timescale (~30 ms), despite neuronal noise and possible jitter accumulation. (2) In accordance with recent experimental findings, the LGN filters out some noise. It thus increases the spike reliability and temporal precision, the sparsity, and, importantly, further decreases down to ~15 ms adjacent cells' correlation timescale. (3) Downstream simple cells in V1's layer 4, if equipped with Spike Timing-Dependent Plasticity (STDP), may detect these fine-scale cross-correlations, and thus connect principally to ON- and OFF-centre cells with Receptive Fields (RF) aligned in the visual space, and thereby become orientation selective, in accordance with Hubel and Wiesel (Journal of Physiology 160:106-154, 1962) classic model. Up to this point we dealt with continuous vision, and there was no absolute time reference such as a stimulus onset, yet information was encoded and decoded in the relative spike times. (4) We then simulated saccades to a static image and benchmarked relative spike time coding and time-to-first spike coding w.r.t. to saccade landing in the context of orientation representation. In both the retina and the LGN, relative spike times are more precise, less affected by pre-landing history and global contrast than absolute ones, and lead to robust contrast invariant orientation representations in V1.

The timing of vision - how neural processing links to different temporal dynamics

In this review, we describe our recent attempts to model the neural correlates of visual perception with biologically inspired networks of spiking neurons, emphasizing the dynamical aspects. Experimental evidence suggests distinct processing modes depending on the type of task the visual system is engaged in. A first mode, crucial for object recognition, deals with rapidly extracting the glimpse of a visual scene in the first 100 ms after its presentation. The promptness of this process points to mainly feedforward processing, which relies on latency coding, and may be shaped by spike timing-dependent plasticity (STDP). Our simulations confirm the plausibility and efficiency of such a scheme. A second mode can be engaged whenever one needs to perform finer perceptual discrimination through evidence accumulation on the order of 400 ms and above. Here, our simulations, together with theoretical considerations, show how predominantly local recurrent connections and long neural time-constants enable the integration and build-up of firing rates on this timescale. In particular, we review how a non-linear model with attractor states induced by strong recurrent connectivity provides straightforward explanations for several recent experimental observations. A third mode, involving additional top-down attentional signals, is relevant for more complex visual scene processing. In the model, as in the brain, these top-down attentional signals shape visual processing by biasing the competition between different pools of neurons. The winning pools may not only have a higher firing rate, but also more synchronous oscillatory activity. This fourth mode, oscillatory activity, leads to faster reaction times and enhanced information transfers in the model. This has indeed been observed experimentally. Moreover, oscillatory activity can format spike times and encode information in the spike phases with respect to the oscillatory cycle. This phenomenon is referred to as “phase-of-firing coding,” and experimental evidence for it is accumulating in the visual system. Simulations show that this code can again be efficiently decoded by STDP. Future work should focus on continuous natural vision, bio-inspired hardware vision systems, and novel experimental paradigms to further distinguish current modeling approaches.

GAD67-GFP+ neurons in the Nucleus of Roller. II. Subthreshold and firing resonance properties

In the companion paper we show that GAD67-GFP+ (GFP+) inhibitory neurons located in the Nucleus of Roller of the mouse brain stem can be classified into two main groups (tonic and phasic) based on their firing patterns in responses to injected depolarizing current steps. In this study we examined the responses of GFP+ cells to fluctuating sinusoidal ("chirp") current stimuli. Membrane impedance profiles in response to chirp stimulation showed that nearly all phasic cells exhibited subthreshold resonance, whereas the majority of tonic GFP+ cells were nonresonant. In general, subthreshold resonance was associated with a relatively fast passive membrane time constant and low input resistance. In response to suprathreshold chirp current stimulation at a holding potential just below spike threshold the majority of tonic GFP+ cells fired multiple action potentials per cycle at low input frequencies (<5 Hz) and either stopped firing or were not entrained by the chirp at higher input frequencies (= tonic low-pass cells). A smaller group of phasic GFP+ cells did not fire at low input frequency but were able to phase-lock 1:1 at intermediate chirp frequencies (= band-pass cells). Spike timing reliability was tested with repeated chirp stimuli and our results show that phasic cells were able to reliably fire when they phase-locked 1:1 over a relatively broad range of input frequencies. Most tonic low-pass cells showed low reliability and poor phase-locking ability. Computer modeling suggested that these different firing resonance properties among GFP+ cells are due to differences in passive and active membrane properties and spiking mechanisms. This heterogeneity of resonance properties might serve to selectively activate subgroups of interneurons.

Enhancement of information transmission of sub-threshold signals applied to distal positions of dendritic trees in hippocampal CA1 neuron models with stochastic resonance

Stochastic resonance (SR) has been shown to enhance the signal-to-noise ratio and detection of low level signals in neurons. It is not yet clear how this effect of SR plays an important role in the information processing of neural networks. The objective of this article is to test the hypothesis that information transmission can be enhanced with SR when sub-threshold signals are applied to distal positions of the dendrites of hippocampal CA1 neuron models. In the computer simulation, random sub-threshold signals were presented repeatedly to a distal position of the main apical branch, while the homogeneous Poisson shot noise was applied as a background noise to the mid-point of a basal dendrite in the CA1 neuron model consisting of the soma with one sodium, one calcium, and five potassium channels. From spike firing times recorded at the soma, the mutual information and information rate of the spike trains were estimated. The simulation results obtained showed a typical resonance curve of SR, and that as the activity (intensity) of sub-threshold signals increased, the maximum value of the information rate tended to increased and eventually SR disappeared. It is concluded that SR can play a key role in enhancing the information transmission of sub-threshold stimuli applied to distal positions on the dendritic trees.

Automatic fitting of spiking neuron models to electrophysiological recordings

Spiking models can accurately predict the spike trains produced by cortical neurons in response to somatically injected currents. Since the specific characteristics of the model depend on the neuron, a computational method is required to fit models to electrophysiological recordings. The fitting procedure can be very time consuming both in terms of computer simulations and in terms of code writing. We present algorithms to fit spiking models to electrophysiological data (time-varying input and spike trains) that can run in parallel on graphics processing units (GPUs). The model fitting library is interfaced with Brian, a neural network simulator in Python. If a GPU is present it uses just-in-time compilation to translate model equations into optimized code. Arbitrary models can then be defined at script level and run on the graphics card. This tool can be used to obtain empirically validated spiking models of neurons in various systems. We demonstrate its use on public data from the INCF Quantitative Single-Neuron Modeling 2009 competition by comparing the performance of a number of neuron spiking models.

Stochastic resonance can enhance information transmission of supra-threshold neural signals

Stochastic resonance (SR) has been shown to improve detection of sub-threshold signals with additive uncor-related background noise, not only in a single hippocampal CA1 neuron model, but in a population of hippocampal CA1 neuron models (Array-Enhanced Stochastic Resonance; AESR). However, most of the information in the CNS is transmitted through supra-threshold signals and the effect of stochastic resonance in neurons on these signals is unknown. Therefore, we investigate through computer simulations whether information transmission of supra-threshold input signal can be improved by uncorrelated noise in a population of hippocampal CA1 neuron models by supra-threshold stochastic resonance (SSR). The mutual information was estimated as an index of information transmission via total and noise entropies from the inter-spike interval (ISI) histograms of the spike trains generated by gathering each of spike trains in a population of hippocampal CA1 neuron models at N = 1, 2, 4, 10, 20 and 50. It was shown that the mutual information was maximized at a specific amplitude of uncorrelated noise, i.e., a typical curve of SR was observed when the number of neurons was greater than 10 with SSR. However, SSR did not affect the information transfer with a small number of neurons. In conclusion, SSR may play an important role in processing information such as memory formation in a population of hippocampal neurons.

Oscillations, phase-of-firing coding, and spike timing-dependent plasticity: an efficient learning scheme

Recent experiments have established that information can be encoded in the spike times of neurons relative to the phase of a background oscillation in the local field potential-a phenomenon referred to as "phase-of-firing coding" (PoFC). These firing phase preferences could result from combining an oscillation in the input current with a stimulus-dependent static component that would produce the variations in preferred phase, but it remains unclear whether these phases are an epiphenomenon or really affect neuronal interactions-only then could they have a functional role. Here we show that PoFC has a major impact on downstream learning and decoding with the now well established spike timing-dependent plasticity (STDP). To be precise, we demonstrate with simulations how a single neuron equipped with STDP robustly detects a pattern of input currents automatically encoded in the phases of a subset of its afferents, and repeating at random intervals. Remarkably, learning is possible even when only a small fraction of the afferents ( approximately 10%) exhibits PoFC. The ability of STDP to detect repeating patterns had been noted before in continuous activity, but it turns out that oscillations greatly facilitate learning. A benchmark with more conventional rate-based codes demonstrates the superiority of oscillations and PoFC for both STDP-based learning and the speed of decoding: the oscillation partially formats the input spike times, so that they mainly depend on the current input currents, and can be efficiently learned by STDP and then recognized in just one oscillation cycle. This suggests a major functional role for oscillatory brain activity that has been widely reported experimentally.

Enhancement of information transmission with stochastic resonance in hippocampal CA1 neuron network

Stochastic resonance (SR) has been shown to improve the detection of subthreshold neural signals in uncorrelated noise. It is yet unclear if and how interactions within a population of neurons can improve information processing in neural networks. In this paper, we investigate the effect of the number of neurons on information transmission in an array of hippocampal CA1 neuron models, i.e., array-enhanced SR (AESR). In computer simulations, the sub-threshold synaptic current (signal) generated by a filtered homogeneous Poisson process was applied to a distal position in each of the apical dendrites, while the background synaptic currents (uncorrelated noise) were presented to a proximal or middle point in each of the dendrites. The transmembrane potentials were recorded at one of the somas in the array of CA1 neuron models, in order to find spike firings and likewise to estimate the total and noise entropies calculated from those spike firing times. The results show that the information rate estimated at the population of the CA1 neuron models is maximized at a specific amplitude of uncorrelated noise, implying AESR. The results further show that the maximum information rate is increased as the number of neurons is increased. It is concluded that AESR can be an important role in information processing is neural systems and that the AESR is modulated by the number of neurons within the network.

The Cauchy problem for one-dimensional spiking neuron models

I consider spiking neuron models defined by a one-dimensional differential equation and a reset-i.e., neuron models of the integrate-and-fire type. I address the question of the existence and uniqueness of a solution on [Formula: see text] for a given initial condition. It turns out that the reset introduces a countable and ordered set of backward solutions for a given initial condition. I discuss the implications of these mathematical results in terms of neural coding and spike timing precision.

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

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

Encoding of information into neural spike trains in an auditory nerve fiber model with electric stimuli in the presence of a pseudospontaneous activity

This paper presents an information-theoretic analysis of neural spike trains in an auditory nerve fiber (ANF) model stimulated extracellularly with Gaussian or sinusoidal waveforms in the presence of a pseudospontaneous activity of spike firings. In the computer simulation, stimulus current waveforms were applied repeatedly to a stimulating electrode located 1 mm above the 26th node of Ranvier, in an ANF axon model having 50 nodes of Ranvier, each consisting of stochastic sodium and potassium channels. From spike firing times recorded at the 36th node of Ranvier, a post-stimulus time histogram (PSTH) was generated, and raster plots were depicted for 30 stimulus presentations, in order to investigate the temporal precision and reliability of the spike firing times. Also, inter spike intervals were generated and then "total" and "noise" entropies were estimated to obtain the mutual information and the information rate of the spike trains. It was shown in the case of Gaussian electric stimuli that the temporal precision of spike firing times and the reliability of spike firings were found to increase as the standard deviation (SD) of the Gaussian electric stimuli increased. It was also shown in the case of sinusoidal electric stimuli where there was a specific amplitude of sinusoidal waveforms, the information rate being maximized. It was implied that setting the parameters of electric stimuli to the specific values which maximize the information rate might contribute to efficiently encoding information into the spike trains in the presence of a pseudospontaneous activity of spike firings.

A benchmark test for a quantitative assessment of simple neuron models

Several methods and algorithms have recently been proposed that allow for the systematic evaluation of simple neuron models from intracellular or extracellular recordings. Models built in this way generate good quantitative predictions of the future activity of neurons under temporally structured current injection. It is, however, difficult to compare the advantages of various models and algorithms since each model is designed for a different set of data. Here, we report about one of the first attempts to establish a benchmark test that permits a systematic comparison of methods and performances in predicting the activity of rat cortical pyramidal neurons. We present early submissions to the benchmark test and discuss implications for the design of future tests and simple neurons models.

Enhancement of information transmission with stochastic resonance in hippocampal CA1 neuron models: effects of noise input location

Stochastic resonance (SR) has been shown to enhance the signal to noise ratio or detection of signals in neurons. It is not yet clear how this effect of SR on the signal to noise ratio affects signal processing in neural networks. In this paper, we investigate the effects of the location of background noise input on information transmission in a hippocampal CA1 neuron model. In the computer simulation, random sub-threshold spike trains (signal) generated by a filtered homogeneous Poisson process were presented repeatedly to the middle point of the main apical branch, while the homogeneous Poisson shot noise (background noise) was applied to a location of the dendrite in the hippocampal CA1 model consisting of the soma with a sodium, a calcium, and five potassium channels. The location of the background noise input was varied along the dendrites to investigate the effects of background noise input location on information transmission. The computer simulation results show that the information rate reached a maximum value for an optimal amplitude of the background noise amplitude. It is also shown that this optimal amplitude of the background noise is independent of the distance between the soma and the noise input location. The results also show that the location of the background noise input does not significantly affect the maximum values of the information rates generated by stochastic resonance.

Enhancement of information transmission with stochastic resonance in hippocampal CA1 neuron models

Stochastic resonance (SR) has been shown to enhance the signal to noise ratio or detection of signals in neurons. It is not yet clear how this effect of SR on the signal to noise ratio affects signal processing in neural networks. In this paper, we test the hypothesis that SR can improve information transmission in the hippocampus. From spike firing times recorded at the soma, the inter spike intervals were generated and then "total" and "noise" entropies were estimated to obtain the mutual information and information rate of the spike trains. The results show that the information rate reached a maximum value at a specific amplitude of the background noise, implying that the stochastic resonance can improve the information transmission in the CA1 neuron model. Furthermore, the results also show that the effect of stochastic resonance tended to decrease as the intensity of the random sub-threshold spike trains (signal) (more than 20 l/s) approached to that of the background noise (100 l/s). In conclusion, the computation results that the stochastic resonance can improve information processing in the hippocampal CA1 neuron model in which the intensity of the random sub-threshold spike trains was set at 5-20 l/s.

Interspike interval statistics in the stochastic Hodgkin-Huxley model: coexistence of gamma frequency bursts and highly irregular firing

When the classical Hodgkin-Huxley equations are simulated with Na- and K-channel noise and constant applied current, the distribution of interspike intervals is bimodal: one part is an exponential tail, as often assumed, while the other is a narrow gaussian peak centered at a short interspike interval value. The gaussian arises from bursts of spikes in the gamma-frequency range, the tail from the interburst intervals, giving overall an extraordinarily high coefficient of variation--up to 2.5 for 180,000 Na channels when I approximately 7 microA/cm(2). Since neurons with a bimodal ISI distribution are common, it may be a useful model for any neuron with class 2 firing. The underlying mechanism is due to a subcritical Hopf bifurcation, together with a switching region in phase-space where a fixed point is very close to a system limit cycle. This mechanism may be present in many different classes of neurons and may contribute to widely observed highly irregular neural spiking.

Information Rate of Neural Spike Trains in Response to Sinusoidal Electric Stimuli in the Presence of a Pseudo-spontaneous Activity

This article presents an analysis of the information rate of neural spike trains in an auditory nerve fiber (ANF) model stimulated extracellularly by sinusoidal electric stimuli under the case where a high-rate pulsatile waveform is presented as a conditioner for increasing the across-fiber-independency, i.e., desynchronization. In the computer simulation, stimulus current waveforms were presented repeatedly to a stimulating electrode located 1 mm above the 26th node of Ranvier, in an ANF axon model having 50 nodes of Ranvier, each consisting of stochastic sodium and potassium channels. From spike firing times recorded at the 36th node of Ranvier, the inter spike intervals were generated and then "total" and "noise" entropies were estimated to obtain the mutual information and information rate of the spike trains. In the present article, it is shown that at a specific amplitude of sinusoidal waveforms, the possibility to encode the sinusoidal function with various amplitudes became greater (enlarging dynamic range), as well as the information rate was found to be maximized. It was implied that setting the amplitude of sinusoids to the specific values which maximize the information rate might contribute to efficiently encoding information under the high-rate pulsatile stimulation in cochlear prostheses.

Information rate of neural spike trains in response to electric stimuli

This article presents an analysis of the information rate of neural spike trains in an auditory nerve fiber (ANF) model stimulated extracellularly by colored Gaussian electric stimuli. In the analysis, stimulus current waveforms were generated by convolving alpha's functions with some alpha's parameters (the inverse of time constants) to white Gaussian processes, and were presented repeatedly to a stimulating electrode located 1 mm above the 26th node of Ranvier, in an ANF axon model having 50 nodes of Ranvier, each consisting of stochastic sodium and potassium channels. From spike firing times recorded at the 36th node of Ranvier, the inter spike intervals were generated and then "total" and "noise" entropies were estimated to obtain the mutual information and information rate of the spike trains. In the present article, it is shown that at a specific alpha parameter, the information rate was found to be maximized. It was implied that setting stimulus parameters to the specific values which maximize the information rate might contribute to efficiently encoding information in neural prostheses.

Exact simulation of integrate-and-fire models with synaptic conductances

Computational neuroscience relies heavily on the simulation of large networks of neuron models. There are essentially two simulation strategies: (1) using an approximation method (e.g., Runge-Kutta) with spike times binned to the time step and (2) calculating spike times exactly in an event-driven fashion. In large networks, the computation time of the best algorithm for either strategy scales linearly with the number of synapses, but each strategy has its own assets and constraints: approximation methods can be applied to any model but are inexact; exact simulation avoids numerical artifacts but is limited to simple models. Previous work has focused on improving the accuracy of approximation methods. In this article, we extend the range of models that can be simulated exactly to a more realistic model: an integrate-and-fire model with exponential synaptic conductances.

Transient resetting: a novel mechanism for synchrony and its biological examples

The study of synchronization in biological systems is essential for the understanding of the rhythmic phenomena of living organisms at both molecular and cellular levels. In this paper, by using simple dynamical systems theory, we present a novel mechanism, named transient resetting, for the synchronization of uncoupled biological oscillators with stimuli. This mechanism not only can unify and extend many existing results on (deterministic and stochastic) stimulus-induced synchrony, but also may actually play an important role in biological rhythms. We argue that transient resetting is a possible mechanism for the synchronization in many biological organisms, which might also be further used in the medical therapy of rhythmic disorders. Examples of the synchronization of neural and circadian oscillators as well as a chaotic neuron model are presented to verify our hypothesis.

Synchronization of dynamical systems is a process whereby two or more systems adjust a given property of their motions to a common behavior due to coupling or forcing. Synchronization has attracted much attention from physicists, biologists, applied mathematicians, and engineers for many years and is a ubiquitous phenomenon. In this paper, Li et al. present a very simple, but general mechanism, called transient resetting, that explains stimulus-induced synchronization in dynamic systems. The mechanism pertains not only to periodic oscillators but also to chaotic ones, and not only to continuous time systems but also to discrete time systems. Biological systems are dynamic, and their synchronization is essential, for example, in the genesis of rhythmic phenomena and information processing. In this paper, the authors study several possible instances of their novel mechanism in a biological context. They also suggest that transient resetting might be used therapeutically in rhythmic disorders. The beneficial role of noise in biological systems has been studied extensively in recent years. Li and colleagues' mechanism provides an explanation for this role in the synchronization in biological systems, even when the stimulus or input to the system is not random or noisy.

Adaptive Exponential Integrate-and-Fire Model as an Effective Description of Neuronal Activity

We introduce a two-dimensional integrate-and-fire model that combines an exponential spike mechanism with an adaptation equation, based on recent theoretical findings. We describe a systematic method to estimate its parameters with simple electrophysiological protocols (current-clamp injection of pulses and ramps) and apply it to a detailed conductance-based model of a regular spiking neuron. Our simple model predicts correctly the timing of 96% of the spikes (±2 ms) of the detailed model in response to injection of noisy synaptic conductances. The model is especially reliable in high-conductance states, typical of cortical activity in vivo, in which intrinsic conductances were found to have a reduced role in shaping spike trains. These results are promising because this simple model has enough expressive power to reproduce qualitatively several electrophysiological classes described in vitro.

Information processing in noisy burster models of sensory neurons

Processing of external stimuli by sensory neurons often involves bursting, when epochs of fast firing alternate with intervals of quiescence. In particular, sensory neurons of electroreceptors in paddlefish (Polyodon spathula) undergo bursting when stimulated externally with broad-band noise, but otherwise fire spontaneously in a quasiperiodic tonic manner. We use a simple phenomenological model for noise-induced bursting to quantify analytically, by means of the Kullback entropy and Fisher information, the gain in information transfer and electroreceptor sensitivity for external noisy stimuli. A good agreement between theoretical predictions, numerical simulations and experimental data is shown.

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.

Generalized Integrate-and-Fire Models of Neuronal Activity Approximate Spike Trains of a Detailed Model to a High Degree of Accuracy

We demonstrate that single-variable integrate-and-fire models can quantitatively capture the dynamics of a physiologically detailed model for fast-spiking cortical neurons. Through a systematic set of approximations, we reduce the conductance-based model to 2 variants of integrate-and-fire models. In the first variant (nonlinear integrate-and-fire model), parameters depend on the instantaneous membrane potential, whereas in the second variant, they depend on the time elapsed since the last spike [Spike Response Model (SRM)]. The direct reduction links features of the simple models to biophysical features of the full conductance-based model. To quantitatively test the predictive power of the SRM and of the nonlinear integrate-and-fire model, we compare spike trains in the simple models to those in the full conductance-based model when the models are subjected to identical randomly fluctuating input. For random current input, the simple models reproduce 70–80 percent of the spikes in the full model (with temporal precision of ±2 ms) over a wide range of firing frequencies. For random conductance injection, up to 73 percent of spikes are coincident. We also present a technique for numerically optimizing parameters in the SRM and the nonlinear integrate-and-fire model based on spike trains in the full conductance-based model. This technique can be used to tune simple models to reproduce spike trains of real neurons.

Phase locking in integrate-and-fire models with refractory periods and modulation

It is known [8, 11, 16, 26] that phase locking can entrain frequency information when the leaky integrate-and-fire (IF) model of a neuron is forced by a periodic function. We show that this is still the case when the IF model is made more biologically realistic. We incorporate into our model spike dependent threshold modulation and refractory periods. Consecutive firing times from this model and their respective interspike intervals are related by an annulus map. We prove a general theorem concerning orientation reversing annulus twist homeomorphisms, which shows that our map admits a unique rotation number. This implies, in particular, that chaotic behaviour is not possible in our model and phase locking is predicted.

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

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

Evaluation of entrainment of a nonlinear neural oscillator to white noise

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

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

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