A biophysical model of synaptic delay learning and temporal pattern recognition in a cerebellar Purkinje cell

Enhanced Release Probability without Changes in Synaptic Delay during Analogue-Digital Facilitation

Neuronal timing with millisecond precision is critical for many brain functions such as sensory perception, learning and memory formation. At the level of the chemical synapse, the synaptic delay is determined by the presynaptic release probability (Pr) and the waveform of the presynaptic action potential (AP). For instance, paired-pulse facilitation or presynaptic long-term potentiation are associated with reductions in the synaptic delay, whereas paired-pulse depression or presynaptic long-term depression are associated with an increased synaptic delay. Parallelly, the AP broadening that results from the inactivation of voltage gated potassium (Kv) channels responsible for the repolarization phase of the AP delays the synaptic response, and the inactivation of sodium (Nav) channels by voltage reduces the synaptic latency. However, whether synaptic delay is modulated during depolarization-induced analogue-digital facilitation (d-ADF), a form of context-dependent synaptic facilitation induced by prolonged depolarization of the presynaptic neuron and mediated by the voltage-inactivation of presynaptic Kv1 channels, remains unclear. We show here that despite Pr being elevated during d-ADF at pyramidal L5-L5 cell synapses, the synaptic delay is surprisingly unchanged. This finding suggests that both Pr- and AP-dependent changes in synaptic delay compensate for each other during d-ADF. We conclude that, in contrast to other short- or long-term modulations of presynaptic release, synaptic timing is not affected during d-ADF because of the opposite interaction of Pr- and AP-dependent modulations of synaptic delay.

A model for time interval learning in the Purkinje cell

Recent experimental findings indicate that Purkinje cells in the cerebellum represent time intervals by mechanisms other than conventional synaptic weights. These findings add to the theoretical and experimental observations suggesting the presence of intra-cellular mechanisms for adaptation and processing. To account for these experimental results we propose a new biophysical model for time interval learning in a Purkinje cell. The numerical model focuses on a classical delay conditioning task (e.g. eyeblink conditioning) and relies on a few computational steps. In particular, the model posits the activation by the parallel fiber input of a local intra-cellular calcium store which can be modulated by intra-cellular pathways. The reciprocal interaction of the calcium signal with several proteins forming negative and positive feedback loops ensures that the timing of inhibition in the Purkinje cell anticipates the interval between parallel and climbing fiber inputs during training. We systematically test the model ability to learn time intervals at the 150-1000 ms time scale, while observing that learning can also extend to the multiple seconds scale. In agreement with experimental observations we also show that the number of pairings required to learn increases with inter-stimulus interval. Finally, we discuss how this model would allow the cerebellum to detect and generate specific spatio-temporal patterns, a classical theory for cerebellar function.

Learned response sequences in cerebellar Purkinje cells

Associative learning in the cerebellum has previously focused on single movements. In eyeblink conditioning, for instance, a subject learns to blink at the right time in response to a conditional stimulus (CS), such as a tone that is repeatedly followed by an unconditional corneal stimulus (US). During conditioning, the CS and US are transmitted by mossy/parallel fibers and climbing fibers to cerebellar Purkinje cells that acquire a precisely timed pause response that drives the overt blink response. The timing of this conditional Purkinje cell response is determined by the CS-US interval and is independent of temporal patterns in the input signal. In addition to single movements, the cerebellum is also believed to be important for learning complex motor programs that require multiple precisely timed muscle contractions, such as, for example, playing the piano. In the present work, we studied Purkinje cells in decerebrate ferrets that were conditioned using electrical stimulation of mossy fiber and climbing fiber afferents as CS and US, while alternating between short and long interstimulus intervals. We found that Purkinje cells can learn double pause responses, separated by an intermediate excitation, where each pause corresponds to one interstimulus interval. The results show that individual cells can not only learn to time a single response but that they also learn an accurately timed sequential response pattern.

Mechanisms for motor timing in the cerebellar cortex

In classical eyeblink conditioning a subject learns to blink to a previously neutral stimulus. This conditional response is timed to occur just before an air puff to the eye. The learning is known to depend on the cerebellar cortex where Purkinje cells respond with adaptively timed pauses in their spontaneous firing. The pauses in the inhibitory Purkinje cells cause disinhibition of the cerebellar nuclei, which elicit the overt blinks. The timing of a Purkinje cell response was previously thought to require a temporal code in the input signal but recent work suggests that the Purkinje cells can learn to time their responses through an intrinsic mechanism that is activated by metabotropic glutamate receptors (mGluR7).

The 40-year history of modeling active dendrites in cerebellar Purkinje cells: emergence of the first single cell "community model"

The subject of the effects of the active properties of the Purkinje cell dendrite on neuronal function has been an active subject of study for more than 40 years. Somewhat unusually, some of these investigations, from the outset have involved an interacting combination of experimental and model-based techniques. This article recounts that 40-year history, and the view of the functional significance of the active properties of the Purkinje cell dendrite that has emerged. It specifically considers the emergence from these efforts of what is arguably the first single cell "community" model in neuroscience. The article also considers the implications of the development of this model for future studies of the complex properties of neuronal dendrites.

Theoretical considerations for understanding a Purkinje cell timing mechanism

In classical conditioning, cerebellar Purkinje cells learn an adaptively timed pause in spontaneous firing. This pause reaches its maximum near the end of the interstimulus interval. While it was thought that this timing was due to temporal patterns in the input signal and selective engagement of changes in synapse strength, we have shown Purkinje cells learn timed responses even when the conditional stimulus is delivered to its immediate afferents.¹ This shows that Purkinje cells have a cellular timing mechanism. The cellular models of intrinsic timing we are aware of are based on adapting the rise time of the concentration of a given ion. As an alternative, we here propose a selection mechanism in abstract terms for how a Purkinje cell could learn to respond at a particular time after an external trigger.

An integrator circuit in cerebellar cortex

The brain builds dynamic models of the body and the outside world to predict the consequences of actions and stimuli. A well-known example is the oculomotor integrator, which anticipates the position-dependent elasticity forces acting on the eye ball by mathematically integrating over time oculomotor velocity commands. Many models of neural integration have been proposed, based on feedback excitation, lateral inhibition or intrinsic neuronal nonlinearities. We report here that a computational model of the cerebellar cortex, a structure thought to implement dynamic models, reveals a hitherto unrecognized integrator circuit. In this model, comprising Purkinje cells, molecular layer interneurons and parallel fibres, Purkinje cells were able to generate responses lasting more than 10 s, to which both neuronal and network mechanisms contributed. Activation of the somatic fast sodium current by subthreshold voltage fluctuations was able to maintain pulse-evoked graded persistent activity, whereas lateral inhibition among Purkinje cells via recurrent axon collaterals further prolonged the responses to step and sine wave stimulation. The responses of Purkinje cells decayed with a time-constant whose value depended on their baseline spike rate, with integration vanishing at low (< 1 per s) and high rates (> 30 per s). The model predicts that the apparently fast circuit of the cerebellar cortex may control the timing of slow processes without having to rely on sensory feedback. Thus, the cerebellar cortex may contain an adaptive temporal integrator, with the sensitivity of integration to the baseline spike rate offering a potential mechanism of plasticity of the response time-constant.

Timing and cue competition in conditioning of the nictitating membrane response of the rabbit (Oryctolagus cuniculus)

Rabbits were classically conditioned using compounds of tone and light conditioned stimuli (CSs) presented with either simultaneous onsets (Experiment 1) or serial onsets (Experiment 2) in a delay conditioning paradigm. Training with the simultaneous compound reduced the likelihood of a conditioned response (CR) to the individual CSs (“mutual overshadowing”) but left CR timing unaltered. CR peaks were consistently clustered around the time of unconditioned stimulus (US) delivery. Training with the serial compound (CSA→CSB→US) reduced responding to CSB (“temporal primacy/information effect”) but this effect was prevented by prior CSB→US pairings. In both cases, serial compound training altered CR timing. On CSA→CSB test trials, the CRs were accelerated; the CR peaks occurred after CSB onset but well before the time of US delivery. Conversely, CRs on CSB– trials were decelerated; the distribution of CR peaks was variable but centered well after the US. Timing on CSB– trials was at most only slightly accelerated. The results are discussed with respect to processes of generalization and spectral timing applicable to the cerebellar and forebrain pathways in eyeblink preparations.

A computational mechanism for unified gain and timing control in the cerebellum

Precise gain and timing control is the goal of cerebellar motor learning. Because the basic neural circuitry of the cerebellum is homogeneous throughout the cerebellar cortex, a single computational mechanism may be used for simultaneous gain and timing control. Although many computational models of the cerebellum have been proposed for either gain or timing control, few models have aimed to unify them. In this paper, we hypothesize that gain and timing control can be unified by learning of the complete waveform of the desired movement profile instructed by climbing fiber signals. To justify our hypothesis, we adopted a large-scale spiking network model of the cerebellum, which was originally developed for cerebellar timing mechanisms to explain the experimental data of Pavlovian delay eyeblink conditioning, to the gain adaptation of optokinetic response (OKR) eye movements. By conducting large-scale computer simulations, we could reproduce some features of OKR adaptation, such as the learning-related change of simple spike firing of model Purkinje cells and vestibular nuclear neurons, simulated gain increase, and frequency-dependent gain increase. These results suggest that the cerebellum may use a single computational mechanism to control gain and timing simultaneously.

Learning stimulus intervals--adaptive timing of conditioned purkinje cell responses

Classical conditioning of motor responses, such as the eyeblink response, is an experimental model of associative learning and of adaptive timing of movements. A conditioned blink will have its maximum amplitude near the expected onset of the unconditioned blink-eliciting stimulus and it adapts to changes in the interval between the conditioned and unconditioned stimuli. Previous studies have shown that an eyeblink conditioning protocol can make cerebellar Purkinje cells learn to pause in response to the conditioned stimulus. According to the cerebellar cortical conditioning model, this conditioned Purkinje cell response drives the overt blink. If so, the model predicts that the temporal properties of the Purkinje cell response reflect the overt behaviour. To test this prediction, in vivo recordings of Purkinje cell activity were performed in decerebrate ferrets during conditioning, using direct stimulation of cerebellar mossy and climbing fibre afferents as conditioned and unconditioned stimuli. The results show that Purkinje cells not only develop a change in responsiveness to the conditioned stimulus. They also learn a particular temporal response profile where the timing, not only of onset and maximum but also of offset, is determined by the temporal interval between the conditioned and unconditioned stimuli.

Postsynaptic signal transduction models for long-term potentiation and depression

More than a hundred biochemical species, activated by neurotransmitters binding to transmembrane receptors, are important in long-term potentiation (LTP) and long-term depression (LTD). To investigate which species and interactions are critical for synaptic plasticity, many computational postsynaptic signal transduction models have been developed. The models range from simple models with a single reversible reaction to detailed models with several hundred kinetic reactions. In this study, more than a hundred models are reviewed, and their features are compared and contrasted so that similarities and differences are more readily apparent. The models are classified according to the type of synaptic plasticity that is modeled (LTP or LTD) and whether they include diffusion or electrophysiological phenomena. Other characteristics that discriminate the models include the phase of synaptic plasticity modeled (induction, expression, or maintenance) and the simulation method used (deterministic or stochastic). We find that models are becoming increasingly sophisticated, by including stochastic properties, integrating with electrophysiological properties of entire neurons, or incorporating diffusion of signaling molecules. Simpler models continue to be developed because they are computationally efficient and allow theoretical analysis. The more complex models permit investigation of mechanisms underlying specific properties and experimental verification of model predictions. Nonetheless, it is difficult to fully comprehend the evolution of these models because (1) several models are not described in detail in the publications, (2) only a few models are provided in existing model databases, and (3) comparison to previous models is lacking. We conclude that the value of these models for understanding molecular mechanisms of synaptic plasticity is increasing and will be enhanced further with more complete descriptions and sharing of the published models.

Timing in trace conditioning of the nictitating membrane response of the rabbit (Oryctolagus cuniculus): scalar, nonscalar, and adaptive features

Using interstimulus intervals (ISIs) of 125, 250, and 500 msec in trace conditioning of the rabbit nictitating membrane response, the offset times and durations of conditioned responses (CRs) were collected along with onset and peak latencies. All measures were proportional to the ISI, but only onset and peak latencies conformed to the criterion for scalar timing. Regarding the CR's possible protective overlap of the unconditioned stimulus (US), CR duration increased with ISI, while the peak's alignment with the US declined. Implications for models of timing and CR adaptiveness are discussed.

The first second: models of short-term memory traces in the brain

Many network models in computational neuroscience rise to the challenge of explaining behavioural phenomena ranging from microseconds to tens of seconds using components operating mostly on a time-scale of milliseconds. These models have in common that the underlying system has a memory, which implies that its output depends on its past input history. In this review we compare how such memory traces or delayed responses may be implemented in different brain areas supporting a diversity of functions.

Computational models of timing mechanisms in the cerebellar granular layer

A long-standing question in neuroscience is how the brain controls movement that requires precisely timed muscle activations. Studies using Pavlovian delay eyeblink conditioning provide good insight into this question. In delay eyeblink conditioning, which is believed to involve the cerebellum, a subject learns an interstimulus interval (ISI) between the onsets of a conditioned stimulus (CS) such as a tone and an unconditioned stimulus such as an airpuff to the eye. After a conditioning phase, the subject’s eyes automatically close or blink when the ISI time has passed after CS onset. This timing information is thought to be represented in some way in the cerebellum. Several computational models of the cerebellum have been proposed to explain the mechanisms of time representation, and they commonly point to the granular layer network. This article will review these computational models and discuss the possible computational power of the cerebellum.

Interaction between Purkinje cells and inhibitory interneurons may create adjustable output waveforms to generate timed cerebellar output

We develop a new model that explains how the cerebellum may generate the timing in classical delay eyeblink conditioning. Recent studies show that both Purkinje cells (PCs) and inhibitory interneurons (INs) have parallel signal processing streams with two time scales: an AMPA receptor-mediated fast process and a metabotropic glutamate receptor (mGluR)-mediated slow process. Moreover, one consistent finding is an increased excitability of PC dendrites (in Larsell's lobule HVI) in animals when they acquire the classical delay eyeblink conditioning naturally, in contrast to in vitro studies, where learning involves long-term depression (LTD). Our model proposes that the delayed response comes from the slow dynamics of mGluR-mediated IP3 activation, and the ensuing calcium concentration change, and not from LTP/LTD. The conditioned stimulus (tone), arriving on the parallel fibers, triggers this slow activation in INs and PC spines. These excitatory (from PC spines) and inhibitory (from INs) signals then interact at the PC dendrites to generate variable waveforms of PC activation. When the unconditioned stimulus (puff), arriving on the climbing fibers, is coupled frequently with this slow activation the waveform is amplified (due to an increased excitability) and leads to a timed pause in the PC population. The disinhibition of deep cerebellar nuclei by this timed pause causes the delayed conditioned response. This suggested PC-IN interaction emphasizes a richer role of the INs in learning and also conforms to the recent evidence that mGluR in the cerebellar cortex may participate in slow motor execution. We show that the suggested mechanism can endow the cerebellar cortex with the versatility to learn almost any temporal pattern, in addition to those that arise in classical conditioning.

Stability of complex spike timing-dependent plasticity in cerebellar learning

Dynamics of spike-timing dependent synaptic plasticity are analyzed for excitatory and inhibitory synapses onto cerebellar Purkinje cells. The purpose of this study is to place theoretical constraints on candidate synaptic learning rules that determine the changes in synaptic efficacy due to pairing complex spikes with presynaptic spikes in parallel fibers and inhibitory interneurons. Constraints are derived for the timing between complex spikes and presynaptic spikes, constraints that result from the stability of the learning dynamics of the learning rule. Potential instabilities in the parallel fiber synaptic learning rule are found to be stabilized by synaptic plasticity at inhibitory synapses if the inhibitory learning rules are stable, and conditions for stability of inhibitory plasticity are given. Combining excitatory with inhibitory plasticity provides a mechanism for minimizing the overall synaptic input. Stable learning rules are shown to be able to sculpt simple-spike patterns by regulating the excitability of neurons in the inferior olive that give rise to climbing fibers.

Generation of time delays: simplified models of intracellular signalling in cerebellar Purkinje cells

In many neuronal systems, information is encoded in temporal spike patterns. The recognition and storage of temporal patterns requires the generation and modulation of time delays between inputs and outputs. In cerebellar Purkinje cells, stimulation of metabotropic glutamate receptors (mGluRs) results in a delayed calcium and voltage response that has been implicated in classical conditioning and temporal pattern recognition. Here, we analyse and simplify a complex model of the intracellular signalling network that has been proposed as a substrate for this delayed response. We systematically simplify the original model, present a minimal model of time delay generation, and show that a delayed response can be produced by the combination of negative feedback and autocatalysis, without any intervening signalling steps that would contribute additive delays. The minimal model is analysed using phase plane methods, and classified as an excitable system. We discuss the implication of excitability for computations performed by intracellular signalling networks in general.

Biophysically detailed modelling of microcircuits and beyond

Realistic bottom-up modelling has been seminal to understanding which properties of microcircuits control their dynamic behaviour, such as the locomotor rhythms generated by central pattern generators. In this article of the TINS Microcircuits Special Feature, we review recent modelling work on the leech-heartbeat and lamprey-swimming pattern generators as examples. Top-down mathematical modelling also has an important role in analyzing microcircuit properties but it has not always been easy to reconcile results from the two modelling approaches. Most realistic microcircuit models are relatively simple and need to be made more detailed to represent complex processes more accurately. We review methods to add neuromechanical feedback, biochemical pathways or full dendritic morphologies to microcircuit models. Finally, we consider the advantages and challenges of full-scale simulation of networks of microcircuits.