Analysis of neural spike trains with interspike interval reconstruction

As a method for the analysis of neural spike trains, we examine fundamental characteristics of interspike interval (ISI) reconstruction theoretically with a leaky-integrator neuron model and experimentally with cricket wind receptor cells. Both the input to the leaky integrator and the stimulus to the wind receptor cells are the time series generated from the Rossler system. By numerical analysis of the leaky integrator, it is shown that, even if ISI reconstruction is possible, sometimes the entire structure of the R?ssler attractor may not be reconstructed with ISI reconstruction. For analysis of the in vivo physiological responses of cricket wind receptor cells, we apply ISI reconstruction, nonlinear prediction and the surrogate data method to the experimental data. As a result of the analysis, it is found that there is a significant deterministic structure in the spike trains. By this analysis of physiological data, it is also shown that, even if ISI reconstruction is possible, the entire attractor may not be reconstructed.     

A proposed neural network for the integrator of the oculomotor system

Single-unit recordings, stimulation studies, and eye movement measurements all indicate that the firing patterns of many oculomotor neurons in the brain stem encode eye-velocity commands in premotor circuits while the firing patterns of extraocular motoneurons contain both eye-velocity and eye-position components. It is necessary to propose that the eye-position component is generated from the eye-velocity signal by a leaky hold element or temporal integrator. Prior models of this integrator suffer from two important problems. Since cells appear to have a steady, background signal when eye position and velocity are zero, how does the integrator avoid integrating this background rate? Most models employ some form of lumped, oositive feedback the gain of which must be kept within totally unreasonable limits for proper operation. We propose a lateral inhibitory network of homogeneous neurons as a model for the neural integrator that solves both problems. Parameter sensitivity studies and lesion simulations are presented to demonstrate robustness of the model with respect to both the choice of parameter values and the consequences of pathological changes in a portion of the neural integrator pool.     

Resonant response of a neural model and of limulus ommatidia to double frequency stimulation

Signals which are the sum of two sinusoids of frequencies v ₁ and v ₂ are used to stimulate: i) an electronic analog of the leaky integrator neural model, ii) the visual neurons of the Limulus lateral eye. This makes it possible to investigate the resonant amplification of the impulse density modulation for v ₁+v ₂ which approaches the free-run discharge rate; this resonance is predicted by the Volterra series representation of the leaky integrator (Poggio and Torre, 1977). The resonant responses obtained look very similar for the simulated discharge and for the experimental one.     

Phase-locked responses in the Limulus lateral eye. Theoretical and experimental investigation.

The 1:1 phase locking of the neural discharge to sinusoidally modulated stimuli was investigated both theoretically and experimentally. On the theoretical side, a neural encoder model, the self-inhibited leaky integrator, was considered, and the phase of the locked impulse was computed for each frequency in the locking range by imposing the condition that the "leaky integral" u(t) of the driving signal should reach the threshold for the first time one stimulus period after the preceding impulse. As u(t) can be a nonmonotonic function, this approach leads to results that sometimes differ from those reported in the literature. It turns out that the phase excursion is often much smaller than the values of about 180 degrees predicted from previous analysis. Moreover, our analysis shows a peculiar effect; the phase locking frequency range narrows when the input modulation depth increases. The theoretical predictions are then compared with phase-locked discharge patterns recorded from visual cells of the Limulus lateral eye, stimulated by sinusoidally modulated light or depolarizing current. The phases of the locked spikes at each of a number of modulation frequencies have been measured. The predictions offered by the model fit the experimental data, although there are some difficulties in determining the effective driving signal.     

The adaptation ability of neuronal models subject to a current step stimulus

Three neuronal models of the spike initiating process were investigated with respect to their ability to show adaptation to a current step: (i) the perfect integrator model (PIM), (ii) the leaky integrator model (LIM), and (iii) the Hodgkin-Huxley (HH-) model. It was found that although each neuronal model will generate different response spike trains to a given stimulus, all responses fulfilled the criteria of a deterministic neural response (Awiszus 1988). The results show that both PIM and LIM are unable to show adaptation regardless of the choice of model parameters whereas the HH-model shows a clear rate of discharge adaptation. The reason for this adaptation lies in the fact that there are conditions for the HH-model where a step stimulus is highly effective. These conditions are investigated by means of a phase plane analysis. Consequences of these results for the explanation of neuronal adaptation and the validity of the neuronal models investigated are discussed.     

Two-neuron networks

The behavior of two pacemaker neurons simulated by leaky integrators and connected reciprocally by synapses was studied. In every case the firing of both neurons phase-locks. The resulting limit cycle may or may not show simultaneous firing of both neurons. When both synapses are excitatory, phase-locking with simultaneous neuronal firing is always present. When one synapse is excitatory and the other inhibitory, phase-locking is also present always, while the neurons may or may not fire simultaneously. For a restricted set of parameters, bistability appears; the initial conditions determine whether or not the limit cycle presents simultaneous firing. When both synapses are inhibitory, the system phase-locks without simultaneous firing for almost every set of parameters.     

Emergence of clusters in the hidden layer of a dynamic recurrent neural network

The neural integrator of the oculomotor system is a privileged field for artificial neural network simulation. In this paper, we were interested in an improvement of the biologically plausible features of the Arnold-Robinson network. This improvement was done by fixing the sign of the connection weights in the network (in order to respect the biological Dale's Law). We also introduced a notion of distance in the network in the form of transmission delays between its units. These modifications necessitated the introduction of a general supervisor in order to train the network to act as a leaky integrator. When examining the lateral connection weights of the hidden layer, the distribution of the weights values was found to exhibit a conspicuous structure: the high-value weights were grouped in what we call clusters. Other zones are quite flat and characterized by low-value weights. Clusters are defined as particular groups of adjoining neurons which have strong and privileged connections with another neighborhood of neurons. The clusters of the trained network are reminiscent of the small clusters or patches that have been found experimentally in the nucleus prepositus hypoglossi, where the neural integrator is located. A study was conducted to determine the conditions of emergence of these clusters in our network: they include the fixation of the weight sign, the introduction of a distance, and a convergence of the information from the hidden layer to the motoneurons. We conclude that this spontaneous emergence of clusters in artificial neural networks, performing a temporal integration, is due to computational constraints, with a restricted space of solutions. Thus, information processing could induce the emergence of iterated patterns in biological neural networks. Received: 18 September 1996 / Accepted in revised form: 7 January 1997     

Analytical reconstruction of the neuronal input current from spike train data

The time course of the current driving action potential generation at a neuron investigated experimentally is in general not measurable directly. In this paper an indirect method is introduced that allows estimation of this unknown current time course using only spike train data. Assuming the leaky integrator model as valid for the action potential encoding site of the investigated neuron, the unknown input current is obtained by determining (analytically) a current time course that upon injection into the leaky integrator model evokes action potential sequences identical to those observed experimentally. Applications of this current-reconstruction procedure to neuronal output data obtained from a leaky integrator model showed that the procedure allows a good estimation of the underlying input current even if the membrane time constant of the investigated neuron is not known exactly. Additionally, an application of current reconstruction to experimental data obtained from a cat muscle spindle primary afferent subject to repeated -stimuli is demonstrated.     

Dynamics of one-dimensional spiking neuron models

In this paper we make a rigorous mathematical analysis of one-dimensional spiking neuron models in a unified framework. We find that, under conditions satisfied in particular by the periodically and aperiodically driven leaky integrator as well as some of its variants, the spike map is increasing on its range, which leaves no room for chaotic behavior. A rigorous expression of the Lyapunov exponent is derived. Finally, we analyse the periodically driven perfect integrator and show that the restriction of the phase map to its range is always conjugated to a rotation, and we provide an explicit expression of the invariant measure.     

Repetitive firing: a quantitative study of feedback in model encoders

Recognition of nonlinearities in the neuronal encoding of repetitive spike trains has generated a number of models to explain this behavior. Here we develop the mathematics and a set of tests for two such models: the leaky integrator and the variable-gamma model. Both of these are nearly sufficient to explain the dynamic behavior of a number of repetitively firing, sensory neurons. Model parameters can be related to possible underlying basic mechanisms. Summed and nonsummed, spike- locked negative feedback are examined in conjunction with the models. Transfer functions are formulated to predict responses to steady state, steps, and sinusoidally varying stimuli in which output data are the times of spike-train events only. An electrical analog model for the leaky integrator is tested to verify predicted responses. Curve fitting and parameter variation techniques are explored for the purpose of extracting basic model parameters from spike train data. Sinusoidal analysis of spike trains appear to be a very accurate method for determining spike-locked feedback parameters, and it is to a large extent a model independent method that may also be applied to neuronal responses.     

Modelling neural informational propagation and functional auditory sensory memory with temporal multi-scale operators

In this paper we prove that both diffusion and the leaky integrators cascade based transport mechanisms have as their inherent property the effect of temporal multi-scaling. The two transport mechanisms are modeled not as convolution based algorithms but as causal physical processes. This implies that propagation of information through a neural map may act as a mechanism for achieving temporal multi-scale analysis in the auditory system. Specifically, we are interested in the effects of such a transport process on the formation and the dynamics of auditory sensory memory. Two temporal models of information propagation are discussed and compared in terms of their ability to model auditory sensory memory effects and the biological plausibility of their structure: the causal diffusion based operator (CD) and the leaky integrator cascade based operator (LINC). We show that temporal multi-scale representations achieved by both models exhibit the effects similar to those of auditory sensory memory (filtering, time delay and binding of information). As regards higher-level functions of auditory sensory memory such as change detection, the LINC operator seems to be a biologically more plausible solution for modeling temporal cortical processing.     

Choice mechanisms for past,temporally extended outcomes

Accurate retrospection is critical in many decision scenarios ranging from investment banking to hedonic psychology. A notoriously difficult case is to integrate previously perceived values over the duration of an experience. Failure in retrospective evaluation leads to suboptimal outcome when previous experiences are under consideration for revisit. A biologically plausible mechanism underlying evaluation of temporally extended outcomes is leaky integration of evidence. The leaky integrator favours positive temporal contrasts, in turn leading to undue emphasis on recency. To investigate choice mechanisms underlying suboptimal outcome based on retrospective evaluation, we used computational and behavioural techniques to model choice between perceived extended outcomes with different temporal profiles. Second-price auctions served to establish the perceived values of virtual coins offered sequentially to humans in a rapid monetary gambling task. Results show that lesser-valued options involving successive growth were systematically preferred to better options with declining temporal profiles. The disadvantageous inclination towards persistent growth was mitigated in some individuals in whom a longer time constant of the leaky integrator resulted in fewer violations of dominance. These results demonstrate how focusing on immediate gains is less beneficial than considering longer perspectives.     

The Frequency Response, Coherence, and Information Capacity of Two Neuronal Models

Two neuronal models are analyzed in which subthreshold inputs are integrated either without loss (perfect integrator) or with a decay which follows an exponential time course (leaky integrator). Linear frequency response functions for these models are compared using sinusoids, Poisson-distributed impulses, or gaussian white noise as inputs. The responses of both models show the nonlinear behavior characteristic of a rectifier for sinusoidal inputs of sufficient amplitude. The leaky integrator shows another nonlinearity in which responses become phase locked to cyclic stimuli. Addition of white noise reduces the distortions due to phase locking. Both models also show selective attenuation of high-frequency components with white noise inputs. Input, output, and cross-spectra are computed using inputs having a broad frequency spectrum. Measures of the coherence and information transmission between the input and output of the models are also derived. Steady inputs, which produce a constant “carrier” rate, and intrinsic sources, which produce variability in the discharge of neurons, may either increase or decrease coherence; however, information transmission using inputs with a broad spectrum is generally increased by steady inputs and reduced by intrinsic variability.     

Functional identification of biological neural networks using reservoir adaptation for point processes

The complexity of biological neural networks does not allow to directly relate their biophysical properties to the dynamics of their electrical activity. We present a reservoir computing approach for functionally identifying a biological neural network, i.e. for building an artificial system that is functionally equivalent to the reference biological network. Employing feed-forward and recurrent networks with fading memory, i.e. reservoirs, we propose a point process based learning algorithm to train the internal parameters of the reservoir and the connectivity between the reservoir and the memoryless readout neurons. Specifically, the model is an Echo State Network (ESN) with leaky integrator neurons, whose individual leakage time constants are also adapted. The proposed ESN algorithm learns a predictive model of stimulus-response relations in in vitro and simulated networks, i.e. it models their response dynamics. Receiver Operating Characteristic (ROC) curve analysis indicates that these ESNs can imitate the response signal of a reference biological network. Reservoir adaptation improved the performance of an ESN over readout-only training methods in many cases. This also held for adaptive feed-forward reservoirs, which had no recurrent dynamics. We demonstrate the predictive power of these ESNs on various tasks with cultured and simulated biological neural networks.     

Accuracy and response-time distributions for decision-making: linear perfect integrators versus nonlinear attractor-based neural circuits

Animals choose actions based on imperfect, ambiguous data. “Noise” inherent in neural processing adds further variability to this already-noisy input signal. Mathematical analysis has suggested that the optimal apparatus (in terms of the speed/accuracy trade-off) for reaching decisions about such noisy inputs is perfect accumulation of the inputs by a temporal integrator. Thus, most highly cited models of neural circuitry underlying decision-making have been instantiations of a perfect integrator. Here, in accordance with a growing mathematical and empirical literature, we describe circumstances in which perfect integration is rendered suboptimal. In particular we highlight the impact of three biological constraints: (1) significant noise arising within the decision-making circuitry itself; (2) bounding of integration by maximal neural firing rates; and (3) time limitations on making a decision. Under conditions (1) and (2), an attractor system with stable attractor states can easily best an integrator when accuracy is more important than speed. Moreover, under conditions in which such stable attractor networks do not best the perfect integrator, a system with unstable initial states can do so if readout of the system’s final state is imperfect. Ubiquitously, an attractor system with a nonselective time-dependent input current is both more accurate and more robust to imprecise tuning of parameters than an integrator with such input. Given that neural responses that switch stochastically between discrete states can “masquerade” as integration in single-neuron and trial-averaged data, our results suggest that such networks should be considered as plausible alternatives to the integrator model.     

Repetitive firing: quantitative analysis of encoder behavior of slowly adapting stretch receptor of crayfish and eccentric cell of Limulus

Techniques developed for determining summed encoder feedback in conjunction with the leaky integrator and variable-gamma models for repetitive firing are applied to spike train data obtained from the slowly adapting crustacean stretch receptor and the eccentric cell of Limulus. Input stimuli were intracellularly applied currents. Analysis of data from cells stringently selected by reproducibility criteria gave a consistent picture for the dynamics of repetitive firing. The variable-gamma model with appropriate summed feedback was most accurate for describing encoding behavior of both cell types. The leaky integrator model, while useful for determining summed feedback parameters, was inadequate to account for underlying mechanisms of encoder activity. For the stretch receptor, two summed feedback processes were detected: one had a short time constant; the other, a long one. Appropriate tests indicated that the short time constant effect was from an electrogenic sodium pump, and the same is presumed for the long time constant summed feedback. Both feedbacks show seasonal and/or species variations. Short hyperpolarizing pulses inhibited the feedback from the long time constant process. The eccentric cell also showed two summed feedback processes: one is due to self inhibition, the other is postulated to be a short time constant electrogenic sodium pump similar to that described in the stretch receptor.     

The effect of a random initial value in neural first-passage-time models

The effect of a random initial value is examined in several stochastic integrate-and-fire neural models with a constant threshold and a constant input. The three models considered are approximations of Stein's model, namely: (1) a leaky integrator with deterministic trajectories, (2) a Wiener process with drift, and (3) an Ornstein-Uhlenbeck process. For model 1, different distributions for the initial value lead to commonly observed interspike interval distributions. For model 2, a discrete and a uniform distribution for the initial value are examined along with some parameter estimation procedures. For model 3, with a truncated normal distribution for the initial value, the coefficient of variation is shown to be greater than 1, and as the threshold becomes large the first-passage-time distribution approaches an exponential distribution. The relationships among the models and between them and previous models are also discussed, along with the robustness of the model assumptions and methods of their verification. The effects of a random initial value are found to be most pronounced at high firing rates.     

Predictive Coding of Dynamical Variables in Balanced Spiking Networks

Two observations about the cortex have puzzled neuroscientists for a long time. First, neural responses are highly variable. Second, the level of excitation and inhibition received by each neuron is tightly balanced at all times. Here, we demonstrate that both properties are necessary consequences of neural networks that represent information efficiently in their spikes. We illustrate this insight with spiking networks that represent dynamical variables. Our approach is based on two assumptions: We assume that information about dynamical variables can be read out linearly from neural spike trains, and we assume that neurons only fire a spike if that improves the representation of the dynamical variables. Based on these assumptions, we derive a network of leaky integrate-and-fire neurons that is able to implement arbitrary linear dynamical systems. We show that the membrane voltage of the neurons is equivalent to a prediction error about a common population-level signal. Among other things, our approach allows us to construct an integrator network of spiking neurons that is robust against many perturbations. Most importantly, neural variability in our networks cannot be equated to noise. Despite exhibiting the same single unit properties as widely used population code models (e.g. tuning curves, Poisson distributed spike trains), balanced networks are orders of magnitudes more reliable. Our approach suggests that spikes do matter when considering how the brain computes, and that the reliability of cortical representations could have been strongly underestimated.     

GPG及其数学研究

本文介绍了中枢模式发生器的数学建模和分析的情况。目前脊椎类生物的中枢模式发生器理论在神经网络研究中起着一个重要的作用。本文介绍了CPG的性质和研究,并回顾了最邻近交联和多重交联的研究情形。特别是考虑了锁相解,频率突跃,停振三种现象。