Temporal correlation based learning in neuron models

We study a learning rule based upon the temporal correlation (weighted by a learning kernel) between incoming spikes and the internal state of the postsynaptic neuron, building upon previous studies of spike timing dependent synaptic plasticity (Kempter, R., Gerstner, W., van Hemmen, J.L., Wagner, H., 1998. Extracting Oscillations: Neuronal coincidence detection with noisy periodic spike input. Neural computation 10, 1987–2017; Kempter, R., Gerstner, W., van Hemmen, J.L., 1999. Hebbian learning and spiking neurons. Physical Reviewm E59, 4498–4514; van Hemmen, J.L., 2001. Theory of synaptic plasticity. In: Moss, F., Gielen, S. (Eds.), Handbook of biological physics. vol. 4, Neuro Informatics, neural modelling, Elsevier, Amsterdam, pp. 771–823. Our learning rule for the synaptic weight w _ij is where the t _j,μ are the arrival times of spikes from the presynaptic neuron j and the function u(t) describes the state of the postsynaptic neuron i. Thus, the spike-triggered average contained in the inner integral is weighted by a kernel Γ(s), the learning window, positive for negative, negative for positive values of the time difference s between post- and presynaptic activity. An antisymmetry assumption for the learning window enables us to derive analytical expressions for a general class of neuron models and to study the changes in input-output relationships following from synaptic weight changes. This is a genuinely non-linear effect (Song, S., Miller, K., Abbott, L., 2000. Competitive Hebbian learning through spike timing dependent synaptic plasticity. Nature Neuroscience 3, 919–926).     

Temporal difference models and reward-related learning in the human brain

Temporal difference learning has been proposed as a model for Pavlovian conditioning, in which an animal learns to predict delivery of reward following presentation of a conditioned stimulus (CS). A key component of this model is a prediction error signal, which, before learning, responds at the time of presentation of reward but, after learning, shifts its response to the time of onset of the CS. In order to test for regions manifesting this signal profile, subjects were scanned using event-related fMRI while undergoing appetitive conditioning with a pleasant taste reward. Regression analyses revealed that responses in ventral striatum and orbitofrontal cortex were significantly correlated with this error signal, suggesting that, during appetitive conditioning, computations described by temporal difference learning are expressed in the human brain.     

Generalized spin models for coupled cortical feature maps obtained by coarse graining correlation based synaptic learning rules

We derive generalized spin models for the development of feedforward cortical architecture from a Hebbian synaptic learning rule in a two layer neural network with nonlinear weight constraints. Our model takes into account the effects of lateral interactions in visual cortex combining local excitation and long range effective inhibition. Our approach allows the principled derivation of developmental rules for low-dimensional feature maps, starting from high-dimensional synaptic learning rules. We incorporate the effects of smooth nonlinear constraints on net synaptic weight projected from units in the thalamic layer (the fan-out) and on the net synaptic weight received by units in the cortical layer (the fan-in). These constraints naturally couple together multiple feature maps such as orientation preference and retinotopic organization. We give a detailed illustration of the method applied to the development of the orientation preference map as a special case, in addition to deriving a model for joint pattern formation in cortical maps of orientation preference, retinotopic location, and receptive field width. We show that the combination of Hebbian learning and center-surround cortical interaction naturally leads to an orientation map development model that is closely related to the XY magnetic lattice model from statistical physics. The results presented here provide justification for phenomenological models studied in Cowan and Friedman (Advances in neural information processing systems 3, 1991), Thomas and Cowan (Phys Rev Lett 92(18):e188101, 2004) and provide a developmental model realizing the synaptic weight constraints previously assumed in Thomas and Cowan (Math Med Biol 23(2):119–138, 2006).     

From spiking neuron models to linear-nonlinear models

Neurons transform time-varying inputs into action potentials emitted stochastically at a time dependent rate. The mapping from current input to output firing rate is often represented with the help of phenomenological models such as the linear-nonlinear (LN) cascade, in which the output firing rate is estimated by applying to the input successively a linear temporal filter and a static non-linear transformation. These simplified models leave out the biophysical details of action potential generation. It is not a priori clear to which extent the input-output mapping of biophysically more realistic, spiking neuron models can be reduced to a simple linear-nonlinear cascade. Here we investigate this question for the leaky integrate-and-fire (LIF), exponential integrate-and-fire (EIF) and conductance-based Wang-Buzsáki models in presence of background synaptic activity. We exploit available analytic results for these models to determine the corresponding linear filter and static non-linearity in a parameter-free form. We show that the obtained functions are identical to the linear filter and static non-linearity determined using standard reverse correlation analysis. We then quantitatively compare the output of the corresponding linear-nonlinear cascade with numerical simulations of spiking neurons, systematically varying the parameters of input signal and background noise. We find that the LN cascade provides accurate estimates of the firing rates of spiking neurons in most of parameter space. For the EIF and Wang-Buzsáki models, we show that the LN cascade can be reduced to a firing rate model, the timescale of which we determine analytically. Finally we introduce an adaptive timescale rate model in which the timescale of the linear filter depends on the instantaneous firing rate. This model leads to highly accurate estimates of instantaneous firing rates.     

Noise-induced transition in excitable neuron models

 We studied the influence of noisy stimulation on the Hodgkin-Huxley neuron model. Rather than examining the noise-related variability of the discharge times of the model – as has been done previously – our study focused on the effect of noise on the stationary distributions of the membrane potential and gating variables of the model. We observed that a gradual increase in the noise intensity did not result in a gradual change of the distributions. Instead, we could identify a critical intermediate noise range in which the shapes of the distributions underwent a drastic qualitative change. Namely, they moved from narrow unimodal Gaussian-like shapes associated with low noise intensities to ones that spread widely at large noise intensities. In particular, for the membrane potential and the sodium activation variable, the distributions changed from unimodal to bimodal. Thus, our investigation revealed a noise-induced transition in the Hodgkin-Huxley model. In order to further characterize this phenomenon, we considered a reduced one-dimensional model of an excitable system, namely the active rotator. For this model, our analysis indicated that the noise-induced transition is associated with a deterministic bifurcation of approximate equations governing the dynamics of the mean and variance of the state variable. Finally, we shed light on the possible functional importance of this noise-induced transition in neuronal coding by determining its effect on the spike timing precision in models of neuronal ensembles. Received: 19 September 2000 / Accepted in revised form: 4 March 2001     

Reduction of conductance-based neuron models

We present a scheme for systematically reducing the number of differential equations required for biophysically realistic neuron models. The techniques are general, are designed to be applicable to a large set of such models and retain in the reduced system as high a degree of fidelity to the original system as possible. As examples, we provide reductions of the Hodgkin-Huxley system and the A-current model of Connor et al. (1977).     

Noise-induced divisive gain control in neuron models

A recent computational study of gain control via shunting inhibition has shown that the slope of the frequency-versus-input (f-I) characteristic of a neuron can be decreased by increasing the noise associated with the inhibitory input (Neural Comput. 13, 227-248). This novel noise-induced divisive gain control relies on the concommittant increase of the noise variance with the mean of the total inhibitory conductance. Here we investigate this effect using different neuronal models. The effect is shown to occur in the standard leaky integrate-and-fire (LIF) model with additive Gaussian white noise, and in the LIF with multiplicative noise acting on the inhibitory conductance. The noisy scaling of input currents is also shown to occur in the one-dimensional theta-neuron model, which has firing dynamics, as well as a large scale compartmental model of a pyramidal cell in the electrosensory lateral line lobe of a weakly electric fish. In this latter case, both the inhibition and the excitatory input have Poisson statistics; noise-induced divisive inhibition is thus seen in f-I curves for which the noise increases along with the input I. We discuss how the variation of the noise intensity along with inputs is constrained by the physiological context and the class of model used, and further provide a comparison of the divisive effect across models.     

Information transmission in multi-input-output stochastic neuron models

The Shannon's information theory in multiway channels (Shannon, 1961) is applied to multi-input-output relations of the stochastic automaton models for interaction of excitatory and inhibitory impulse sequences proposed in the previous papers (Tsukada et al., 1977). In these models, the output spike train depends upon several statistical characteristics (mean frequency, standard deviation, form, order-dependence or order-independence, etc.) of the excitatory and inhibitory input spike trains. By the use of the multiple-access channel in information theory, some stochastic properties of temporal pattern discrimination in neurons are analyzed and discussed with biological systems.     

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.     

Temporal discrimination learning by pigeons

Memory for time by animals appears to undergo a systematic shortening. This so-called choose-short effect can be seen in a conditional temporal discrimination when a delay is inserted between the sample and comparison stimuli. We have proposed that this temporal shortening may result from a procedural artifact in which the delay appears similar to the intertrial interval and thus, produces an inadvertent ambiguity or 'instructional failure'. When this ambiguity is avoided by distinguishing the intertrial interval from the delay, as well as the samples from the delay, the temporal shortening effect and other asymmetries often disappear. By avoiding artifacts that can lead to a misinterpretation of results, we may understand better how animals represent time. An alternative procedure for studying temporal discriminations is with the psychophysical bisection procedure in which following conditional discrimination training, intermediate durations are presented and the point of subjective equality is determined. Research using the bisection procedure has shown that pigeons represent temporal durations not only as their absolute value but also relative to durations from which they must be discriminated. Using this procedure, we have also found that time passes subjectively slower when animals are required to respond to the to-be-timed stimulus.     

Event-based minimum-time control of oscillatory neuron models

We present an event-based feedback control method for randomizing the asymptotic phase of oscillatory neurons. Phase randomization is achieved by driving the neuron’s state to its phaseless set, a point at which its phase is undefined and is extremely sensitive to background noise. We consider the biologically relevant case of a fixed magnitude constraint on the stimulus signal, and show how the control objective can be accomplished in minimum time. The control synthesis problem is addressed using the minimum-time-optimal Hamilton–Jacobi–Bellman framework, which is quite general and can be applied to any spiking neuron model in the conductance-based Hodgkin–Huxley formalism. We also use this methodology to compute a feedback control protocol for optimal spike rate increase. This framework provides a straightforward means of visualizing isochrons, without actually calculating them in the traditional way. Finally, we present an extension of the phase randomizing control scheme that is applied at the population level, to a network of globally coupled neurons that are firing in synchrony. The applied control signal desynchronizes the population in a demand-controlled way.     

Animal models for motor neuron disease.

Motor neuron disease is a general term applied to a broad class of neurodegenerative diseases that are characterized by fatally progressive muscular weakness, atrophy, and paralysis attributable to loss of motor neurons. At present, there is no cure for most motor neuron diseases, including amyotrophic lateral sclerosis (ALS), the most common human motor neuron disease--the cause of which remains largely unknown. Animal models of motor neuron disease (MND) have significantly contributed to the remarkable recent progress in understanding the cause, genetic factors, and pathologic mechanisms proposed for this class of human neurodegenerative disorders. Largely driven by ALS research, animal models of MND have proven their usefulness in elucidating potential causes and specific pathogenic mechanisms, and have helped to advance promising new treatments from "benchside to bedside." This review summarizes important features of selected established animal models of MND: genetically engineered mice and inherited or spontaneously occurring MND in the murine, canine, and equine species.     

Effective stimuli for constructing reliable neuron models

The rich dynamical nature of neurons poses major conceptual and technical challenges for unraveling their nonlinear membrane properties. Traditionally, various current waveforms have been injected at the soma to probe neuron dynamics, but the rationale for selecting specific stimuli has never been rigorously justified. The present experimental and theoretical study proposes a novel framework, inspired by learning theory, for objectively selecting the stimuli that best unravel the neuron's dynamics. The efficacy of stimuli is assessed in terms of their ability to constrain the parameter space of biophysically detailed conductance-based models that faithfully replicate the neuron's dynamics as attested by their ability to generalize well to the neuron's response to novel experimental stimuli. We used this framework to evaluate a variety of stimuli in different types of cortical neurons, ages and animals. Despite their simplicity, a set of stimuli consisting of step and ramp current pulses outperforms synaptic-like noisy stimuli in revealing the dynamics of these neurons. The general framework that we propose paves a new way for defining, evaluating and standardizing effective electrical probing of neurons and will thus lay the foundation for a much deeper understanding of the electrical nature of these highly sophisticated and non-linear devices and of the neuronal networks that they compose.     

A Volterra representation for some neuron models

A Volterra-like polynomial representation is derived and its convergence discussed for two neuronal models in which subthreshold inputs are integrated either without loss (integrate and fire) or with a decay which follows an exponential time course (leaky integrator). This polynomial representation provides a kind of nonlinear transfer function for the nonlinear encoding process. Standard formulae are used to derive explicitely the output for various inputs as in linear system theory. Moreover, the nonlinear transfer function associated with cascades or networks of neurons can be also obtained. Finally, extensions and implications of these results are discussed.     

Pathways to motor neuron degeneration in transgenic mouse models

Amyotrophic lateral sclerosis (ALS) is an adult-onset neurological disorder characterized by the selective loss of motor neurons. A pathological hallmark of both sporadic and familial ALS is the presence of abnormal accumulations of neurofilament and peripherin proteins in motor neurons. In the past decade, transgenic mouse approaches have been used to address the role of such cytoskeletal abnormalities in motor neuron disease and also to unravel the pathogenesis caused by mutations in the gene coding for superoxide dismutase 1 (SOD1) that account for ~20% of familial ALS cases. In mouse models, disparate effects could result from different types of intermediate filament (IF) aggregates. Perikaryal IF accumulations induced by the overexpression of any of the three wild-type neurofilament proteins were quite well tolerated by motor neurons. Indeed, perikaryal swellings provoked by NF-H overexpression can even confer protection against toxicity of mutant SOD1. Other types of IF aggregates seem neurotoxic, such as those found in transgenic mice overexpressing either peripherin or an assembly-disrupting NF-L mutant. Moreover, understanding the toxicity of SOD1 mutations has been surprisingly difficult. The analysis of transgenic mice expressing mutant SOD1 has yielded complex results, suggesting that multiple pathways may contribute to disease that include the involvement of non-neuronal cells.     

System identification based on point processes and correlation densities. I. The nonrefractory neuron model

This paper discusses a neuron model that transforms an input point process into an output point process. The model is composed of two stages: a linear filter operates on the input process, whereafter an instantaneous nonlinearity generates the output point process. If the pulse generator is exponential, it is possible to derive formulas that explicitly express the output autocorrelation and input-output cross-correlation densities into the connectivity function and input autocorrelation densities. Alternatively these formulas may be used to derive connectivity from correlation densities. This identification method is tested on computer simulations of the model.     

A role for phasic dopamine neuron firing in habit learning

In this issue of Neuron, Wang et?al. (2011) show that mice with dopamine neuron-specific NMDAR1 deletion have attenuated phasic dopamine neuron firing and a deficit in habit learning. These findings indicate that brain regions sensitive to phasic dopamine signals may underlie habit learning.     

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 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.