Hybrid spiking models.

I review a class of hybrid models of neurons that combine continuous spike-generation mechanisms and a discontinuous 'after-spike' reset of state variables. Unlike Hodgkin-Huxley-type conductance-based models, the hybrid spiking models have a few parameters derived from the bifurcation theory; instead of matching neuronal electrophysiology, they match neuronal dynamics. I present a method of after-spike resetting suitable for hardware implementation of such models, and a hybrid numerical method for simulations of large-scale biological spiking networks.

Spike-timing theory of working memory.

Working memory (WM) is the part of the brain's memory system that provides temporary storage and manipulation of information necessary for cognition. Although WM has limited capacity at any given time, it has vast memory content in the sense that it acts on the brain's nearly infinite repertoire of lifetime long-term memories. Using simulations, we show that large memory content and WM functionality emerge spontaneously if we take the spike-timing nature of neuronal processing into account. Here, memories are represented by extensively overlapping groups of neurons that exhibit stereotypical time-locked spatiotemporal spike-timing patterns, called polychronous patterns; and synapses forming such polychronous neuronal groups (PNGs) are subject to associative synaptic plasticity in the form of both long-term and short-term spike-timing dependent plasticity. While long-term potentiation is essential in PNG formation, we show how short-term plasticity can temporarily strengthen the synapses of selected PNGs and lead to an increase in the spontaneous reactivation rate of these PNGs. This increased reactivation rate, consistent with in vivo recordings during WM tasks, results in high interspike interval variability and irregular, yet systematically changing, elevated firing rate profiles within the neurons of the selected PNGs. Additionally, our theory explains the relationship between such slowly changing firing rates and precisely timed spikes, and it reveals a novel relationship between WM and the perception of time on the order of seconds.

Large-scale model of mammalian thalamocortical systems.

The understanding of the structural and dynamic complexity of mammalian brains is greatly facilitated by computer simulations. We present here a detailed large-scale thalamocortical model based on experimental measures in several mammalian species. The model spans three anatomical scales. (i) It is based on global (white-matter) thalamocortical anatomy obtained by means of diffusion tensor imaging (DTI) of a human brain. (ii) It includes multiple thalamic nuclei and six-layered cortical microcircuitry based on in vitro labeling and three-dimensional reconstruction of single neurons of cat visual cortex. (iii) It has 22 basic types of neurons with appropriate laminar distribution of their branching dendritic trees. The model simulates one million multicompartmental spiking neurons calibrated to reproduce known types of responses recorded in vitro in rats. It has almost half a billion synapses with appropriate receptor kinetics, short-term plasticity, and long-term dendritic spike-timing-dependent synaptic plasticity (dendritic STDP). The model exhibits behavioral regimes of normal brain activity that were not explicitly built-in but emerged spontaneously as the result of interactions among anatomical and dynamic processes. We describe spontaneous activity, sensitivity to changes in individual neurons, emergence of waves and rhythms, and functional connectivity on different scales.

Solving the distal reward problem through linkage of STDP and dopamine signaling.

In Pavlovian and instrumental conditioning, reward typically comes seconds after reward-triggering actions, creating an explanatory conundrum known as "distal reward problem": How does the brain know what firing patterns of what neurons are responsible for the reward if 1) the patterns are no longer there when the reward arrives and 2) all neurons and synapses are active during the waiting period to the reward? Here, we show how the conundrum is resolved by a model network of cortical spiking neurons with spike-timing-dependent plasticity (STDP) modulated by dopamine (DA). Although STDP is triggered by nearly coincident firing patterns on a millisecond timescale, slow kinetics of subsequent synaptic plasticity is sensitive to changes in the extracellular DA concentration during the critical period of a few seconds. Random firings during the waiting period to the reward do not affect STDP and hence make the network insensitive to the ongoing activity-the key feature that distinguishes our approach from previous theoretical studies, which implicitly assume that the network be quiet during the waiting period or that the patterns be preserved until the reward arrives. This study emphasizes the importance of precise firing patterns in brain dynamics and suggests how a global diffusive reinforcement signal in the form of extracellular DA can selectively influence the right synapses at the right time.

Polychronization: computation with spikes.

We present a minimal spiking network that can polychronize, that is, exhibit reproducible time-locked but not synchronous firing patterns with millisecond precision, as in synfire braids. The network consists of cortical spiking neurons with axonal conduction delays and spike-timing-dependent plasticity (STDP); a ready-to-use MATLAB code is included. It exhibits sleeplike oscillations, gamma (40 Hz) rhythms, conversion of firing rates to spike timings, and other interesting regimes. Due to the interplay between the delays and STDP, the spiking neurons spontaneously self-organize into groups and generate patterns of stereotypical polychronous activity. To our surprise, the number of coexisting polychronous groups far exceeds the number of neurons in the network, resulting in an unprecedented memory capacity of the system. We speculate on the significance of polychrony to the theory of neuronal group selection (TNGS, neural Darwinism), cognitive neural computations, binding and gamma rhythm, mechanisms of attention, and consciousness as "attention to memories."

Which model to use for cortical spiking neurons?

We discuss the biological plausibility and computational efficiency of some of the most useful models of spiking and bursting neurons. We compare their applicability to large-scale simulations of cortical neural networks.

Spike-timing dynamics of neuronal groups.

A neuronal network inspired by the anatomy of the cerebral cortex was simulated to study the self-organization of spiking neurons into neuronal groups. The network consisted of 100 000 reentrantly interconnected neurons exhibiting known types of cortical firing patterns, receptor kinetics, short-term plasticity and long-term spike-timing-dependent plasticity (STDP), as well as a distribution of axonal conduction delays. The dynamics of the network allowed us to study the fine temporal structure of emerging firing patterns with millisecond resolution. We found that the interplay between STDP and conduction delays gave rise to the spontaneous formation of neuronal groups--sets of strongly connected neurons capable of firing time-locked, although not necessarily synchronous, spikes. Despite the noise present in the model, such groups repeatedly generated patterns of activity with millisecond spike-timing precision. Exploration of the model allowed us to characterize various group properties, including spatial distribution, size, growth, rate of birth, lifespan, and persistence in the presence of synaptic turnover. Localized coherent input resulted in shifts of receptive and projective fields in the model similar to those observed in vivo.

Relating STDP to BCM.

We demonstrate that the BCM learning rule follows directly from STDP when pre- and postsynaptic neurons fire uncorrelated or weakly correlated Poisson spike trains, and only nearest-neighbor spike interactions are taken into account.

Bursts as a unit of neural information: selective communication via resonance.

What is the functional significance of generating a burst of spikes, as opposed to a single spike? A dominant point of view is that bursts are needed to increase the reliability of communication between neurons. Here, we discuss the alternative, but complementary, hypothesis: bursts with specific resonant interspike frequencies are more likely to cause a postsynaptic cell to fire than are bursts with higher or lower frequencies. Such a frequency preference might occur at the level of individual synapses because of the interplay between short-term synaptic depression and facilitation, or at the postsynaptic cell level because of subthreshold membrane potential oscillations and resonance. As a result, the same burst could resonate for some synapses or cells and not resonate for others, depending on their natural resonance frequencies. This observation suggests that, in addition to increasing reliability of synaptic transmission, bursts of action potentials might provide effective mechanisms for selective communication between neurons.

Resonance and selective communication via bursts in neurons having subthreshold oscillations.

Revealing the role of bursts of action potentials is an important step toward understanding how the neurons communicate. The dominant point of view is that bursts are needed to increase the reliability of communication between neurons [Trends Neurosci. 20 (1997) 38]. In this paper we present an alternative but complementary hypothesis. We consider the effect of a short burst on a model postsynaptic cell having damped oscillation of its membrane potential. The oscillation frequency (eigenfrequency) plays a crucial role. Due to the subthreshold membrane resonance and frequency preference, the responses (i.e. voltage oscillations) of such a cell are amplified when the intra-burst frequency equals the cell's eigenfrequency. Responses are negligible, however, if the intra-burst frequency is twice the eigenfrequency. Thus, the same burst could be effective for one cell and ineffective for another depending on their eigenfrequencies. This theoretical observation suggests that, in addition to coping with unreliable synapses, bursts of action potentials may provide effective mechanisms for selective communication between neurons.

Multiple cusp bifurcations.

The cusp bifurcation provides one of the simplest routes leading to bistability and hysteresis in neuron dynamics. We show that weakly connected networks of neurons near cusp bifurcations that satisfy a certain adaptation condition have quite interesting and complicated dynamics. First, we prove that any such network can be transformed into a canonical model by an appropriate continuous change of variables. Then we show that the canonical model can operate as a multiple attractor neural network or as a globally asymptotically stable neural network depending on the choice of parameters.