Simon R. Schultz

The Role of Spike Timing in the Coding of Stimulus Location in Rat Somatosensory Cortex

Although the timing of single spikes is known to code for time-varying features of a sensory stimulus, it remains unclear whether time is also exploited in the neuronal coding of the spatial structure of the environment, where nontemporal stimulus features are fundamental. This report demonstrates that, in the whisker representation of rat cortex, precise spike timing of single neurons increases the information transmitted about stimulus location by 44%, compared to that transmitted only by the total number of spikes. Crucial to this code is the timing of the first spike after whisker movement. Complex, single neuron spike patterns play a smaller, synergistic role. Timing permits very few spikes to transmit high quantities of information about a behaviorally significant, spatial stimulus.

Correlations and the encoding of information in the nervous system

Is the information transmitted by an ensemble of neurons determined solely by the number of spikes fired by each cell, or do correlations in the emission of action potentials also play a significant role? We derive a simple formula which enables this question to be answered rigorously for short time–scales. The formula quantifies the corrections to the instantaneous information rate which result from correlations in spike emission between pairs of neurons. The mutual information that the ensemble of neurons conveys about external stimuli can thus be broken down into firing rate and correlation components. This analysis provides fundamental constraints upon the nature of information coding, showing that over short time–scales, correlations cannot dominate information representation, that stimulus–independent correlations may lead to synergy (where the neurons together convey more information than they would if they were considered independently), but that only certain combinations of the different sources of correlation result in significant synergy rather than in redundancy or in negligible effects. This analysis leads to a new quantification procedure which is directly applicable to simultaneous multiple neuron recordings.

A Unified Approach to the Study of Temporal, Correlational, and Rate Coding

We demonstrate that the information contained in the spike occurrence times of a population of neurons can be broken up into a series of terms, each reflecting something about potential coding mechanisms. This is possible in the coding regime in which few spikes are emitted in the relevant time window. This approach allows us to study the additional information contributed by spike timing beyond that present in the spike counts and to examine the contributions to the whole information of different statistical properties of spike trains, such as firing rates and correlation functions. It thus forms the basis for a new quantitative procedure for analyzing simultaneous multiple neuron recordings and provides theoretical constraints on neural coding strategies. We find a transition between two coding regimes, depending on the size of the relevant observation timescale. For time windows shorter than the timescale of the stimulus-induced response fluctuations, there exists a spike count coding phase, in which the purely temporal information is of third order in time. For time windows much longer than the characteristic timescale, there can be additional timing information of first order, leading to a temporal coding phase in which timing information may affect the instantaneous information rate. In this new framework, we study the relative contributions of the dynamic firing rate and correlation variables to the full temporal information, the interaction of signal and noise correlations in temporal coding, synergy between spikes and between cells, and the effect of refractoriness. We illustrate the utility of the technique by analyzing a few cells from the rat barrel cortex.

Spatial Pattern Coding of Sensory Information by Climbing Fiber-Evoked Calcium Signals in Networks of Neighboring Cerebellar Purkinje Cells

Climbing fiber input produces complex spike synchrony across populations of cerebellar Purkinje cells oriented in the parasagittal axis. Elucidating the fine spatial structure of this synchrony is crucial for understanding its role in the encoding and processing of sensory information within the olivocerebellar cortical circuit. We investigated these issues using in vivo multineuron two-photon calcium imaging in combination with information theoretic analysis. Spontaneous dendritic calcium transients linked to climbing fiber input were observed in multiple neighboring Purkinje cells. Spontaneous synchrony of calcium transients between individual Purkinje cells falls off over ∼200 μm mediolaterally, consistent with the presence of cerebellar microzones organized by climbing fiber input. Synchrony was increased after administration of harmaline, consistent with an olivary origin. Periodic sensory stimulation also resulted in a transient increase of synchrony after stimulus onset. To examine how synchrony affects the neural population code provided by the spatial pattern of complex spikes, we analyzed its information content. We found that spatial patterns of calcium events from small ensembles of cells provided substantially more stimulus information (59% more for seven-cell ensembles) than available by counting events across the pool without taking into account spatial origin. Information theoretic analysis indicated that, rather than contributing significantly to sensory coding via stimulus dependence, correlational effects on sensory coding are dominated by redundancy attributable to the prevalent spontaneous synchrony. The olivocerebellar circuit thus uses a labeled line code to report sensory signals, leaving open a role for synchrony in flexible selection of signals for output to deep cerebellar nuclei.

On decoding the responses of a population of neurons from short time windows

The effectiveness of various stimulus identification (decoding) procedures for extracting the information carried by the responses of a population of neurons to a set of repeatedly presented stimuli is studied analytically, in the limit of short time windows. It is shown that in this limit, the entire information content of the responses can sometimes be decoded, and when this is not the case, the lost information is quantified. In particular, the mutual information extracted by taking into account only the most likely stimulus in each trial turns out to be, if not equal, much closer to the true value than that calculated from all the probabilities that each of the possible stimuli in the set was the actual one. The relation between the mutual information extracted by decoding and the percentage of correct stimulus decodings is also derived analytically in the same limit, showing that the metric content index can be estimated reliably from a few cells recorded from brief periods. Computer simulations as well as the activity of real neurons recorded in the primate hippocampus serve to confirm these results and illustrate the utility and limitations of the approach.

The role of correlations in direction and contrast coding in the primary visual cortex.

The spiking activity of nearby cortical neurons is not independent. Numerous studies have explored the importance of this correlated responsivity for visual coding and perception, often by comparing the information conveyed by pairs of simultaneously recorded neurons with the sum of information provided by the respective individual cells. Pairwise responses typically provide slightly more information so that encoding is weakly synergistic. The simple comparison between pairwise and summed individual responses conflates several forms of correlation, however, making it impossible to judge the relative importance of synchronous spiking, basic tuning properties, and stimulus-independent and stimulus-dependent correlation. We have applied an information theoretic approach to this question, using the responses of pairs of neurons to drifting sinusoidal gratings of different directions and contrasts that have been recorded in the primary visual cortex of anesthetized macaque monkeys. Our approach allows us to break down the information provided by pairs of neurons into a number of components. This analysis reveals that, although synchrony is prevalent and informative, the additional information it provides frequently is offset by the redundancy arising from the similar tuning properties of the two cells. Thus coding is approximately independent with weak synergy or redundancy arising, depending on the similarity in tuning and the temporal precision of the analysis. We suggest that this would allow cortical circuits to enjoy the stability provided by having similarly tuned neurons without suffering the penalty of redundancy, because the associated information transmission deficit is compensated for by stimulus-dependent synchrony.

A finite rate of innovation algorithm for fast and accurate spike detection from two-photon calcium imaging

OBJECTIVE Inferring the times of sequences of action potentials (APs) (spike trains) from neurophysiological data is a key problem in computational neuroscience. The detection of APs from two-photon imaging of calcium signals offers certain advantages over traditional electrophysiological approaches, as up to thousands of spatially and immunohistochemically defined neurons can be recorded simultaneously. However, due to noise, dye buffering and the limited sampling rates in common microscopy configurations, accurate detection of APs from calcium time series has proved to be a difficult problem. APPROACH Here we introduce a novel approach to the problem making use of finite rate of innovation (FRI) theory (Vetterli et al 2002 IEEE Trans. SIGNAL PROCESS: 50 1417-28). For calcium transients well fit by a single exponential, the problem is reduced to reconstructing a stream of decaying exponentials. Signals made of a combination of exponentially decaying functions with different onset times are a subclass of FRI signals, for which much theory has recently been developed by the signal processing community. Main results. We demonstrate for the first time the use of FRI theory to retrieve the timing of APs from calcium transient time series. The final algorithm is fast, non-iterative and parallelizable. Spike inference can be performed in real-time for a population of neurons and does not require any training phase or learning to initialize parameters. SIGNIFICANCE The algorithm has been tested with both real data (obtained by simultaneous electrophysiology and multiphoton imaging of calcium signals in cerebellar Purkinje cell dendrites), and surrogate data, and outperforms several recently proposed methods for spike train inference from calcium imaging data.

Analysis of Information Transmission in the Schaffer Collaterals

Hippocampal region CA1 seems from comparative studies to be particularly important in the primate brain, in addition to being crucial to memory function. Thus, it is an extremely appropriate place to begin a quantitative investigation of the information representation and transmission capabilities of cerebral neural networks. In this study, a mathematical model of the Schaffer collateral projection from CA3 to CA1 is described. From the model, the amount of information that can be conveyed by the Schaffer collaterals is calculated, i.e., the information that a pattern of firing in CA1 conveys about a pattern of firing in CA3, because of the connections between them. The calculation is performed analytically for an arbitrary probability distribution describing the pattern of CA3 firing and then solved numerically for particular input distributions. The effect of a number of issues on the information conveyed is examined. Consideration of the effect of the amount of analog resolution of firing rates in the patterns of activity in CA3 confirmed information transmission to be most efficient for binary codes, to a degree that depends on the sparseness of activity. For very sparse codes, a binary code allows more information to be received even in absolute terms, but for more distributed codes, slightly more information can be received by CA1 by making use of analog resolution. The pattern of convergence of connections from CA3 to CA1 is examined, i.e., the spatial distribution of the number of connections each CA1 neuron receives. It is found that the effect of the difference between a uniform convergence model and a proposed real convergence pattern (Bernard and Wheal, Hippocampus 1994;4:497–529) is minimal. The effect of the ratio of expansion between CA3 and CA1 due to the relative numbers of neurons in these two areas is studied. The Schaffer collaterals in all mammalian species reported in the literature seem to operate in a régime in which there is at least the scope for efficient transfer of information. In addition, the effect of topography (with respect to the transverse hippocampal axis) in the Schaffer collateral connectivity is examined. In the absence of spatial correlations, topography is found to have essentially no effect on information transmission. If spatial correlations in firing were present in CA3 (which, however, would be less efficient for memory storage in the recurrent collaterals), information transmission would be maximized by matching the topographic spread to the spatial scale of correlation. Hippocampus 1999;9:582–598.

The Ising decoder: reading out the activity of large neural ensembles

The Ising model has recently received much attention for the statistical description of neural spike train data. In this paper, we propose and demonstrate its use for building decoders capable of predicting, on a millisecond timescale, the stimulus represented by a pattern of neural activity. After fitting to a training dataset, the Ising decoder can be applied “online” for instantaneous decoding of test data. While such models can be fit exactly using Boltzmann learning, this approach rapidly becomes computationally intractable as neural ensemble size increases. We show that several approaches, including the Thouless–Anderson–Palmer (TAP) mean field approach from statistical physics, and the recently developed Minimum Probability Flow Learning (MPFL) algorithm, can be used for rapid inference of model parameters in large-scale neural ensembles. Use of the Ising model for decoding, unlike other problems such as functional connectivity estimation, requires estimation of the partition function. As this involves summation over all possible responses, this step can be limiting. Mean field approaches avoid this problem by providing an analytical expression for the partition function. We demonstrate these decoding techniques by applying them to simulated neural ensemble responses from a mouse visual cortex model, finding an improvement in decoder performance for a model with heterogeneous as opposed to homogeneous neural tuning and response properties. Our results demonstrate the practicality of using the Ising model to read out, or decode, spatial patterns of activity comprised of many hundreds of neurons.

A large-scale spiking neural network accelerator for FPGA systems

Spiking neural networks (SNN) aim to mimic membrane potential dynamics of biological neurons. They have been used widely in neuromorphic applications and neuroscience modeling studies. We design a parallel SNN accelerator for producing large-scale cortical simulation targeting an off-the-shelf Field-Programmable Gate Array (FPGA)-based system. The accelerator parallelizes synaptic processing with run time proportional to the firing rate of the network. Using only one FPGA, this accelerator is estimated to support simulation of 64K neurons 2.5 times real-time, and achieves a spike delivery rate which is at least 1.4 times faster than a recent GPU accelerator with a benchmark toroidal network.