Werner M. Kistler

Mathematical Formulations of Hebbian Learning

Abstract. Several formulations of correlation-based Hebbian learning are reviewed. On the presynaptic side, activity is described either by a firing rate or by presynaptic spike arrival. The state of the postsynaptic neuron can be described by its membrane potential, its firing rate, or the timing of backpropagating action potentials (BPAPs). It is shown that all of the above formulations can be derived from the point of view of an expansion. In the absence of BPAPs, it is natural to correlate presynaptic spikes with the postsynaptic membrane potential. Time windows of spike-time-dependent plasticity arise naturally if the timing of postsynaptic spikes is available at the site of the synapse, as is the case in the presence of BPAPs. With an appropriate choice of parameters, Hebbian synaptic plasticity has intrinsic normalization properties that stabilizes postsynaptic firing rates and leads to subtractive weight normalization.

Targeted mutation of Cyln2 in the Williams syndrome critical region links CLIP-115 haploinsufficiency to neurodevelopmental abnormalities in mice

Williams syndrome is a neurodevelopmental disorder caused by the hemizygous deletion of 1.6 Mb on human chromosome 7q11.23. This region comprises the gene CYLN2, encoding CLIP-115, a microtubule-binding protein of 115 kD. Using a gene-targeting approach, we provide evidence that mice with haploinsufficiency for Cyln2 have features reminiscent of Williams syndrome, including mild growth deficiency, brain abnormalities, hippocampal dysfunction and particular deficits in motor coordination. Absence of CLIP-115 also leads to increased levels of CLIP-170 (a closely related cytoplasmic linker protein) and dynactin at the tips of growing microtubules. This protein redistribution may affect dynein motor regulation and, together with the loss of CLIP-115–specific functions, underlie neurological alterations in Williams syndrome.

Analysis of Cx36 knockout does not support tenet that olivary gap junctions are required for complex spike synchronization and normal motor performance

Abstract: Electrotonic coupling by gap junctions between neurons in the inferior olive has been claimed to underly complex spike (CS) synchrony of Purkinje cells in the cerebellar cortex and thereby to play a role in the coordination of movements. Here, we investigated the motor performance of mice that lack connexin36 (Cx36), which appears necessary for functional olivary gap junctions. Cx36 null‐mutants are not ataxic, they show a normal performance on the accelerating rotorod, and they have a regular walking pattern. In addition, they show normal compensatory eye movements during sinusoidal visual and/or vestibular stimulation. To find out whether the normal motor performance in mutants reflects normal CS activity or some compensatory mechanism downstream of the cerebellar cortex, we determined the CS firing rate, climbing‐fiber pause, and degree of CS synchrony. None of these parameters in the mutants differed from those in wildtype littermates. Finally, we investigated whether the role of coupling becomes apparent under challenging conditions, such as during application of the tremorgenic drug harmaline, which specifically turns olivary neurons into an oscillatory state at a high frequency. In both the mutants and wildtypes this application induced tremors of a similar duration with similar peak frequencies and amplitudes. Thus surprisingly, the present data does not support the notion that electrotonic coupling by gap junctions underlies synchronization of olivary spike activity and that these gap junctions are essential for normal motor performance.

Time windows and reverberating loops: a reverse-engineering approach to cerebellar function.

We review a reverse-engineering approach to cerebellar function that pays particular attention to temporal aspects of neuronal interactions. This approach offers new vistas on the role of GABAergic synapses and reverberating projections within the olivocerebellar system. More specifically, our simulations show that Golgi cells can control the ring time of granule cells rather than their ring rate and that Purkinje cells can trigger precisely timed rebound spikes in neurons of the deep cerebellar nuclei. This rebound activity can reverberate back to the cerebellar cortex giving rise to a complex oscillatory dynamics that may have interesting functional implications for working memory and timed-response tasks.

Dynamical working memory and timed responses: the role of reverberating loops in the olivo-cerebellar system

This article explores dynamical properties of the olivo-cerebellar system that arise from the specific wiring of inferior olive (IO), cerebellar cortex, and deep cerebellar nuclei (DCN). We show that the irregularity observed in the firing pattern of the IO neurons is not necessarily produced by noise but can instead be the result of a purely deterministic network effect. We propose that this effect can serve as a dynamical working memory or as a neuronal clock with a characteristic timescale of about 100 ms that is determined by the slow calcium dynamics of IO and DCN neurons. This concept provides a novel explanation of how the cerebellum can solve timing tasks on a timescale that is two orders of magnitude longer than the millisecond timescale usually attributed to neuronal dynamics. One of the key ingredients of our model is the observation that due to postinhibitory rebound, DCN neurons can be driven by GABAergic (inhibitory) input from cerebellar Purkinje cells. Topographic projections from the DCN to the IO form a closed reverberating loop with an overall synaptic transmission delay of about 100 ms that is in resonance with the intrinsic oscillatory properties of the inferior olive. We use a simple time-discrete model based on McCulloch-Pitts neurons in order to investigate in a first step some of the fundamental properties of a network with delayed reverberating projections. The macroscopic behavior is analyzed by means of a mean-field approximation. Numerical simulations, however, show that the microscopic dynamics has a surprisingly rich structure that does not show up in a mean-field description. We have thus performed extensive numerical experiments in order to quantify the ability of the network to serve as a dynamical working memory and its vulnerability by noise. In a second step, we develop a more realistic conductance-based network model of the inferior olive consisting of about 20 multicompartment neurons that are coupled by gap junctions and receive excitatory and inhibitory synaptic input via AMPA and GABAergic synapses. The simulations show that results for the time-discrete model hold true in a time-continuous description.

Spike-timing dependent synaptic plasticity: A phenomenological framework

Abstract. In this paper a phenomenological model of spike-timing dependent synaptic plasticity (STDP) is developed that is based on a Volterra series-like expansion. Synaptic weight changes as a function of the relative timing of pre- and postsynaptic spikes are described by integral kernels that can easily be inferred from experimental data. The resulting weight dynamics can be stated in terms of statistical properties of pre- and postsynaptic spike trains. Generalizations to neurons that fire two different types of action potentials, such as cerebellar Purkinje cells where synaptic plasticity depends on correlations in two distinct presynaptic fibers, are discussed.We show that synaptic plasticity, together with strictly local bounds for the weights, can result in synaptic competition that is required for any form of pattern formation. This is illustrated by a concrete example where a single neuron equipped with STDP can selectively strengthen those synapses with presynaptic neurons that reliably deliver precisely timed spikes at the expense of other synapses which transmit spikes with a broad temporal distribution. Such a mechanism may be of vital importance for any neuronal system where information is coded in the timing of individual action potentials.

Introduction: neurons and mathematics

The primary aim of this chapter is to introduce several elementary notions of neuroscience, in particular the concepts of action potentials, postsynaptic potentials, firing thresholds, refractoriness, and adaptation. Based on these notions a preliminary model of neuronal dynamics is built and this simple model (the leaky integrate-and-fire model) will be used as a starting point and reference for the generalized integrate-and-fire models, which are the main topic of the book, to be discussed in Parts II and III. Since the mathematics used for the simple model is essentially that of a one-dimensional linear differential equation, we take this first chapter as an opportunity to introduce some of the mathematical notation that will be used throughout the rest of the book. Owing to the limitations of space, we cannot – and do not want to – give a comprehensive introduction to such a complex field as neurobiology. The presentation of the biological background in this chapter is therefore highly selective and focuses on those aspects needed to appreciate the biological background of the theoretical work presented in this book. For an in-depth discussion of neurobiology we refer the reader to the literature mentioned at the end of this chapter. After the review of neuronal properties in Sections 1.1 and 1.2 we will turn, in Section 1.3, to our first mathematical neuron model. The last two sections are devoted to a discussion of the strengths and limitations of simplified models.

Nonlinear integrate-and-fire models

Detailed conductance-based neuron models can reproduce electrophysiological measurements to a high degree of accuracy, but because of their intrinsic complexity these models are difficult to analyze. For this reason, simple phenomenological spiking neuron models are highly popular for studies of neural coding, memory, and network dynamics. In this chapter we discuss formal threshold models of neuronal firing, also called integrate-andfire models. The shape of the action potential of a given neuron is rather stereotyped with very little change between one spike and the next. Thus, the shape of the action potential which travels along the axon to a postsynaptic neuron cannot be used to transmit information; rather, from the point of view of the receiving neuron, action potentials are “events” which are fully characterized by the arrival time of the spike at the synapse. Note that spikes from different neuron types can have different shapes and the duration and shape of the spike does influence neurotransmitter release; but the spikes that arrive at a given synapse all come from the same presynaptic neuron and – if we neglect effects of fatigue of ionic channels in the axon – we can assume that its time course is always the same. Therefore we make no effort to model the exact shape of an action potential. Rather, spikes are treated as events characterized by their firing time – and the task consists in finding a model so as to reliably predict spike timings.