Nonlinear principal components analysis of neuronal spike train data

Many recent approaches to decoding neural spike trains depend critically on the assumption that for low-pass filtered spike trains, the temporal structure is optimally represented by a small number of linear projections onto the data. We therefore tested this assumption of linearity by comparing a linear factor analysis technique (principal components analysis) with a nonlinear neural network based method. It is first shown that the nonlinear technique can reliably identify a neuronally plausible nonlinearity in synthetic spike trains. However, when applied to the outputs from primary visual cortical neurons, this method shows no evidence for significant temporal nonlinearities. The implications of this are discussed. Received: 29 November 1996 / Accepted in revised form: 1 July 1997     

Quantification and statistical verification of neuronal stimulus responses from noisy spike train data

Usually neuronal responses to short-lasting stimuli are displayed as peri-stimulus time histogram. The function estimated by such a histogram allows to obtain informations about stimulus-induced postsynaptic events as long as the interpretation is restricted to the first response component after the stimulus. The interpretation of secondary response components is much more difficult, as they may be either due to stimulus effects or represent an echo of the primary response. In the present paper two output functions are developed that do not show such an echoing of responses. The first one, the interspike interval change function, represents an ideal way to quantify a neuronal stimulus response as its amplitude was found to be almost independent of the stimulation strategy used during acquisition of the spike train data. The other function, the displaced impulses function, allows to verify the statistical significance of an observed response component. Both functions may be estimated from stimulus-correlated spike train data, even if the neuron under investigation shows considerable interspike-interval variability in the absence of stimulation. The concepts underlying these neuronal output functions are developed on simulated responses of a Hodgkin-Huxley-type model for a mammalian neuron at body temperature that is exposed to a transient excitatory conductance increase. Additionally, estimation of these output functions is also demonstrated on responses of human soleus motoneurons that were exposed to electrical stimuli of the tibial nerve in the popliteal fossa.     

Fourier analysis of spike train data

A formal representation of nerve spike trains in the form of a sum of rectangular functions is proposed. This formal instantaneous frequency function can be Fourier analyzed. The resulting algorithm has the useful properties of spike by spike calculations and an insensitivity to the mean (carrier) spike rate. The technique is also useful for producing a smooth (filtered) reconstruction of a spike train.     

Causal pattern recovery from neural spike train data using the Snap Shot Score

We present a new approach to learning directed information flow networks from multi-channel spike train data. A novel scoring function, the Snap Shot Score, is used to assess potential networks with respect to their quality of causal explanation for the data. Additionally, we suggest a generic concept of plausibility in order to assess network learning techniques under partial observability conditions. Examples demonstrate the assessment of networks with the Snap Shot Score, and neural network simulations show its performance in complex situations with partial observability. We discuss the application of the new score to real data and indicate how it can be modified to suit other neural data types.     

The stochastic properties of input spike trains control neuronal arithmetic

In the nervous system, the representation of signals is based predominantly on the rate and timing of neuronal discharges. In most everyday tasks, the brain has to carry out a variety of mathematical operations on the discharge patterns. Recent findings show that even single neurons are capable of performing basic arithmetic on the sequences of spikes. However, the interaction of the two spike trains, and thus the resulting arithmetic operation may be influenced by the stochastic properties of the interacting spike trains. If we represent the individual discharges as events of a random point process, then an arithmetical operation is given by the interaction of two point processes. Employing a probabilistic model based on detection of coincidence of random events and complementary computer simulations, we show that the point process statistics control the arithmetical operation being performed and, particularly, that it is possible to switch from subtraction to division solely by changing the distribution of the inter-event intervals of the processes. Consequences of the model for evaluation of binaural information in the auditory brainstem are demonstrated. The results accentuate the importance of the stochastic properties of neuronal discharge patterns for information processing in the brain; further studies related to neuronal arithmetic should therefore consider the statistics of the interacting spike trains.     

State-space analysis of time-varying higher-order spike correlation for multiple neural spike train data

Precise spike coordination between the spiking activities of multiple neurons is suggested as an indication of coordinated network activity in active cell assemblies. Spike correlation analysis aims to identify such cooperative network activity by detecting excess spike synchrony in simultaneously recorded multiple neural spike sequences. Cooperative activity is expected to organize dynamically during behavior and cognition; therefore currently available analysis techniques must be extended to enable the estimation of multiple time-varying spike interactions between neurons simultaneously. In particular, new methods must take advantage of the simultaneous observations of multiple neurons by addressing their higher-order dependencies, which cannot be revealed by pairwise analyses alone. In this paper, we develop a method for estimating time-varying spike interactions by means of a state-space analysis. Discretized parallel spike sequences are modeled as multi-variate binary processes using a log-linear model that provides a well-defined measure of higher-order spike correlation in an information geometry framework. We construct a recursive Bayesian filter/smoother for the extraction of spike interaction parameters. This method can simultaneously estimate the dynamic pairwise spike interactions of multiple single neurons, thereby extending the Ising/spin-glass model analysis of multiple neural spike train data to a nonstationary analysis. Furthermore, the method can estimate dynamic higher-order spike interactions. To validate the inclusion of the higher-order terms in the model, we construct an approximation method to assess the goodness-of-fit to spike data. In addition, we formulate a test method for the presence of higher-order spike correlation even in nonstationary spike data, e.g., data from awake behaving animals. The utility of the proposed methods is tested using simulated spike data with known underlying correlation dynamics. Finally, we apply the methods to neural spike data simultaneously recorded from the motor cortex of an awake monkey and demonstrate that the higher-order spike correlation organizes dynamically in relation to a behavioral demand.     

A method of neuronal spike train study based on autoregressive analysis

The method of autoregressive (AR) analysis for neuronal spike trains (NST) is proposed in the paper. The AR model and the Green's function as well as the power spectral density function are used to process and analyse the neuronal interspike interval (ISI) sequences of cat's first somatosensory area of cortex (SI area) under various situations. With these methods the characteristics of the ISI sequence such as the AR order and parameters, memory property, correlativity and periodicity etc. can be extracted.     

Combining frequency and time domain approaches to systems with multiple spike train input and output

A frequency domain approach and a time domain approach have been combined in an investigation of the behaviour of the primary and secondary endings of an isolated muscle spindle in response to the activity of two static fusimotor axons when the parent muscle is held at a fixed length and when it is subjected to random length changes. The frequency domain analysis has an associated error process which provides a measure of how well the input processes can be used to predict the output processes and is also used to specify how the interactions between the recorded processes contribute to this error. Without assuming stationarity of the input, the time domain approach uses a sequence of probability models of increasing complexity in which the number of input processes to the model is progressively increased. This feature of the time domain approach was used to identify a preferred direction of interaction between the processes underlying the generation of the activity of the primary and secondary endings. In the presence of fusimotor activity and dynamic length changes imposed on the muscle, it was shown that the activity of the primary and secondary endings carried different information about the effects of the inputs imposed on the muscle spindle. The results presented in this work emphasise that the analysis of the behaviour of complex systems benefits from a combination of frequency and time domain methods. This article is part of a special issue on Neuronal Dynamics of Sensory Coding.     

Contrasting effects of the persistent Na+ current on neuronal excitability and spike timing

The persistent Na+ current, INaP, is known to amplify subthreshold oscillations and synaptic potentials, but its impact on action potential generation remains enigmatic. Using computational modeling, whole-cell recording, and dynamic clamp of CA1 hippocampal pyramidal cells in brain slices, we examined how INaP changes the transduction of excitatory current into action potentials. Model simulations predicted that INaP increases afterhyperpolarizations, and, although it increases excitability by reducing rheobase, INaP also reduces the gain in discharge frequency in response to depolarizing current (f/I gain). These predictions were experimentally confirmed by using dynamic clamp, thus circumventing the longstanding problem that INaP cannot be selectively blocked. Furthermore, we found that INaP increased firing regularity in response to sustained depolarization, although it decreased spike time precision in response to single evoked EPSPs. Finally, model simulations demonstrated that I(NaP) increased the relative refractory period and decreased interspike-interval variability under conditions resembling an active network in vivo.     

Sensitivity of different stimulus-timing strategies for the detection of small excitations in noisy spike train data

There are several different strategies to control the timing of a stimulus with respect to the ongoing discharge during the recording of neuronal stimulus-response characteristics. One possible strategy consists of delivering stimuli in such a way that a constant pre-stimulus spike density is reached. Another strategy enforces spike application with a constant stimulus latency after a spontaneous discharge. In this paper the sensitivity of these different strategies for statistical verification of small excitatory response components was investigated. It was found that the difference between observed poststimulus spike distribution and expected spike distribution under the null hypothesis of no stimulus effect was larger using a constant-stimulus-latency (CSL) strategy with an appropriate value for the stimulus latency. Thus, the statistical verification of neuronal response components is clearly facilitated if a CSL strategy is used. This superiority of the CSL strategy is marked, especially for small excitations at neurons discharging slowly with low discharge variability.     

Pooling of impulse sequences,with emphasis on applications to neuronal spike data

Summary A mathematical model is presented that is supposed to describe those types of neuronal discharges which show a preponderance of short intervals, as well as one or more preferred intervals of a longer duration. It is assumed that via two channels impulses impinge upon a nerve cell and that each impulse gives rise to a response. The intervals between impulses in one channel are distributed according to an exponential, or an exponential-like, function; those in the other channel are distributed according to a monomodal, or a multimodal, function.The interval distributions and the expectation density (auto-correlation) functions of the model are in particular compared with data on thalamic neuron discharge patterns reported in the literature.The properties of superimposed time series of events would seem to be of a wider interest, stretching beyond the field of theoretical neurophysiology. It is indicated how the theory is of use in the detection of hidden rhythms in records which are composed of a mixture of different signals.     

Alias-free sampling of neuronal spike trains

Summary Spectral analysis provides powerful techniques for describing the lower order moments of a stochastic process and interactions between two or more stochastic processes. A major problem in the application of spectral analysis to neuronal spike trains is how to obtain equispaced samples of the spike trains which will give unbiased and alias-free spectral estimates. Various sampling methods, which treat the spike train as a continuous signal, a point process and as a series of Dirac delta-functions, are reviewed and their limitations discussed. A new sampling technique, which gives unbiased and alias-free estimates, is described. This technique treats the spike train as a series of delta functions and generates samples by digital filtering. Implementation of this technique on a small computer is simple and virtually on-line.     

Recurrence plots of neuronal spike trains

The recently developed qualitative method of diagnosis of dynamical systems — recurrence plots has been applied to the analysis of dynamics of neuronal spike trains recorded from cerebellum and red nucleus of anesthetized cats. Recurrence plots revealed robust and common changes in the similarity structure of interspike interval sequences as well as significant deviations from randomness in serial ordering of intervals. Recurring episodes of alike, quasi-deterministic firing patterns suggest the spontaneous modulation of the dynamical complexity of the trajectories of observed neurons. These modulations are associated with changing dynamical properties of a neuronal spike-train-generating system. Their existence is compatible with the information processing paradigm of attractor neural networks.     

A cross-interval spike train analysis: the correlation between spike generation and temporal integration of doublets

A stochastic spike train analysis technique is introduced to reveal the correlation between the firing of the next spike and the temporal integration period of two consecutive spikes (i.e., a doublet). Statistics of spike firing times between neurons are established to obtain the conditional probability of spike firing in relation to the integration period. The existence of a temporal integration period is deduced from the time interval between two consecutive spikes fired in a reference neuron as a precondition to the generation of the next spike in a compared neuron. This analysis can show whether the coupled spike firing in the compared neuron is correlated with the last or the second-to-last spike in the reference neuron. Analysis of simulated and experimentally recorded biological spike trains shows that the effects of excitatory and inhibitory temporal integration are extracted by this method without relying on any subthreshold potential recordings. The analysis also shows that, with temporal integration, a neuron driven by random firing patterns can produce fairly regular firing patterns under appropriate conditions. This regularity in firing can be enhanced by temporal integration of spikes in a chain of polysynaptically connected neurons. The bandpass filtering of spike firings by temporal integration is discussed. The results also reveal that signal transmission delays may be attributed not just to conduction and synaptic delays, but also to the delay time needed for temporal integration. Received: 3 March 1997 / Accepted in revised form: 6 November 1997     

Balanced synaptic input shapes the correlation between neural spike trains

Stimulus properties, attention, and behavioral context influence correlations between the spike times produced by a pair of neurons. However, the biophysical mechanisms that modulate these correlations are poorly understood. With a combined theoretical and experimental approach, we show that the rate of balanced excitatory and inhibitory synaptic input modulates the magnitude and timescale of pairwise spike train correlation. High rate synaptic inputs promote spike time synchrony rather than long timescale spike rate correlations, while low rate synaptic inputs produce opposite results. This correlation shaping is due to a combination of enhanced high frequency input transfer and reduced firing rate gain in the high input rate state compared to the low state. Our study extends neural modulation from single neuron responses to population activity, a necessary step in understanding how the dynamics and processing of neural activity change across distinct brain states.     

Spectral analysis of input spike trains by spike-timing-dependent plasticity

Spike-timing-dependent plasticity (STDP) has been observed in many brain areas such as sensory cortices, where it is hypothesized to structure synaptic connections between neurons. Previous studies have demonstrated how STDP can capture spiking information at short timescales using specific input configurations, such as coincident spiking, spike patterns and oscillatory spike trains. However, the corresponding computation in the case of arbitrary input signals is still unclear. This paper provides an overarching picture of the algorithm inherent to STDP, tying together many previous results for commonly used models of pairwise STDP. For a single neuron with plastic excitatory synapses, we show how STDP performs a spectral analysis on the temporal cross-correlograms between its afferent spike trains. The postsynaptic responses and STDP learning window determine kernel functions that specify how the neuron "sees" the input correlations. We thus denote this unsupervised learning scheme as 'kernel spectral component analysis' (kSCA). In particular, the whole input correlation structure must be considered since all plastic synapses compete with each other. We find that kSCA is enhanced when weight-dependent STDP induces gradual synaptic competition. For a spiking neuron with a "linear" response and pairwise STDP alone, we find that kSCA resembles principal component analysis (PCA). However, plain STDP does not isolate correlation sources in general, e.g., when they are mixed among the input spike trains. In other words, it does not perform independent component analysis (ICA). Tuning the neuron to a single correlation source can be achieved when STDP is paired with a homeostatic mechanism that reinforces the competition between synaptic inputs. Our results suggest that neuronal networks equipped with STDP can process signals encoded in the transient spiking activity at the timescales of tens of milliseconds for usual STDP.     

Conductance versus current noise in a neuronal model for noisy subthreshold oscillations and related spike generation

Biological systems are notoriously noisy. Noise, therefore, also plays an important role in many models of neural impulse generation. Noise is not only introduced for more realistic simulations but also to account for cooperative effects between noisy and nonlinear dynamics. Often, this is achieved by a simple noise term in the membrane equation (current noise). However, there are ongoing discussions whether such current noise is justified or whether rather conductance noise should be introduced because it is closer to the natural origin of noise. Therefore, we have compared the effects of current and conductance noise in a neuronal model for subthreshold oscillations and action potential generation. We did not see any significant differences in the model behavior with respect to voltage traces, tuning curves of interspike intervals, interval distributions or frequency responses when the noise strength is adjusted. These findings indicate that simple current noise can give reasonable results in neuronal simulations with regard to physiological relevant noise effects.