Mikhail I. Rabinovich

Synchronous Behavior of Two Coupled Biological Neurons

We report experimental studies of synchronization phenomena in a pair of biological neurons that interact through naturally occurring, electrical coupling. When these neurons generate irregular bursts of spikes, the natural coupling synchronizes slow oscillations of membrane potential, but not the fast spikes. By adding artificial electrical coupling we studied transitions between synchrony and asynchrony in both slow oscillations and fast spikes. We discuss the dynamics of bursting and synchronization in living neurons with distributed functional morphology. [S0031-9007(98)08008-9] The dynamics of many neural ensembles such as central pattern generators (CPGs) or thalamo-cortical circuits pose questions related to cooperative behavior of neurons. Individual neurons may show irregular behavior [1], while ensembles of different neurons can synchronize in order to process biological information [2] or to produce regular, rhythmical activity [3]. How do the irregular neurons synchronize? How do they inhibit noise and intrinsic fluctuations? What parameters of the ensemble are responsible for such synchronization and regularization? Answers to these and similar questions may be found through experiments that enable one to follow qualitatively the cooperative dynamics of neurons as intrinsic and synaptic parameters are varied. Despite their interest, these problems have not received extensive study. Results of such an experiment for a minimal ensemble of two coupled, living neurons are reported in this communication. The experiment was carried out on two electrically coupled neurons (the pyloric dilators, PD) from the pyloric CPG of the lobster stomatogastric ganglion [3]. Individually, these neurons can generate spiking-bursting activity that is irregular and seemingly chaotic. This activity pattern can be altered by injecting dc current (I1 and I2) into the neurons; see Fig. 1. In parallel to their natural coupling, we added artificial coupling by a dynamic current clamp device [7]. Varying these control parameters (offset current and artificial coupling), we found the following regimes of cooperative behavior. Natural coupling produces state-dependent synchronization; see Fig. 2. (i) When depolarized by positive dc current, both neurons fire a continuous pattern of synchronized spikes (Fig. 2d). (ii) With little or no applied current, the neurons fire spikes in irregular bursts: now the slow oscillations are well synchronized while spikes are not (Fig. 2a). Changing the magnitude and sign of electrical coupling restructures the cooperative dynamics. (iii) Increasing the strength of coupling produces complete synchronization of both irregular slow oscillations and fast spikes (see below). (iv) Compensating the natural coupling leads to the onset

Transient Cognitive Dynamics, Metastability, and Decision Making

The idea that cognitive activity can be understood using nonlinear dynamics has been intensively discussed at length for the last 15 years. One of the popular points of view is that metastable states play a key role in the execution of cognitive functions. Experimental and modeling studies suggest that most of these functions are the result of transient activity of large-scale brain networks in the presence of noise. Such transients may consist of a sequential switching between different metastable cognitive states. The main problem faced when using dynamical theory to describe transient cognitive processes is the fundamental contradiction between reproducibility and flexibility of transient behavior. In this paper, we propose a theoretical description of transient cognitive dynamics based on the interaction of functionally dependent metastable cognitive states. The mathematical image of such transient activity is a stable heteroclinic channel, i.e., a set of trajectories in the vicinity of a heteroclinic skeleton that consists of saddles and unstable separatrices that connect their surroundings. We suggest a basic mathematical model, a strongly dissipative dynamical system, and formulate the conditions for the robustness and reproducibility of cognitive transients that satisfy the competing requirements for stability and flexibility. Based on this approach, we describe here an effective solution for the problem of sequential decision making, represented as a fixed time game: a player takes sequential actions in a changing noisy environment so as to maximize a cumulative reward. As we predict and verify in computer simulations, noise plays an important role in optimizing the gain.

Dynamical Encoding by Networks of Competing Neuron Groups

Following studies of olfactory processing in insects and fish, we investigate neural networks whose dynamics in phase space is represented by orbits near the heteroclinic connections between saddle regions (fixed points or limit cycles). These networks encode input information as trajectories along the heteroclinic connections. If there are N neurons in the network, the capacity is approximately e(N-1)!, i.e., much larger than that of most traditional network structures. We show that a small winnerless competition network composed of FitzHugh-Nagumo spiking neurons efficiently transforms input information into a spatiotemporal output.

Model of Transient Oscillatory Synchronization in the Locust Antennal Lobe

Transient pairwise synchronization of locust antennal lobe (AL) projection neurons (PNs) occurs during odor responses. In a Hodgkin-Huxley-type model of the AL, interactions between excitatory PNs and inhibitory local neurons (LNs) created coherent network oscillations during odor stimulation. GABAergic interconnections between LNs led to competition among them such that different groups of LNs oscillated with periodic Ca(2+) spikes during different 50-250 ms temporal epochs, similar to those recorded in vivo. During these epochs, LN-evoked IPSPs caused phase-locked, population oscillations in sets of postsynaptic PNs. The model shows how alternations of the inhibitory drive can temporally encode sensory information in networks of neurons without precisely tuned intrinsic oscillatory properties.

Model of Cellular and Network Mechanisms for Odor-Evoked Temporal Patterning in the Locust Antennal Lobe

Locust antennal lobe (AL) projection neurons (PNs) respond to olfactory stimuli with sequences of depolarizing and hyperpolarizing epochs, each lasting hundreds of milliseconds. A computer simulation of an AL network was used to test the hypothesis that slow inhibitory connections between local neurons (LNs) and PNs are responsible for temporal patterning. Activation of slow inhibitory receptors on PNs by the same GABAergic synapses that underlie fast oscillatory synchronization of PNs was sufficient to shape slow response modulations. This slow stimulus- and neuron-specific patterning of AL activity was resistant to blockade of fast inhibition. Fast and slow inhibitory mechanisms at synapses between LNs and PNs can thus form dynamical PN assemblies whose elements synchronize transiently and oscillate collectively, as observed not only in the locust AL, but also in the vertebrate olfactory bulb.

Synchronized action of synaptically coupled chaotic model neurons

Experimental observations of the intracellular recorded electrical activity in individual neurons show that the temporal behavior is often chaotic. We discuss both our own observations on a cell from the stom-atogastric central pattern generator of lobster and earlier observations in other cells. In this paper we work with models of chaotic neurons, building on models by Hindmarsh and Rose for bursting, spiking activity in neurons. The key feature of these simplified models of neurons is the presence of coupled slow and fast subsystems. We analyze the model neurons using the same tools employed in the analysis of our experimental data. We couple two model neurons both electrotonically and electrochemically in inhibitory and excitatory fashions. In each of these cases, we demonstrate that the model neurons can synchronize in phase and out of phase depending on the strength of the coupling. For normal synaptic coupling, we have a time delay between the action of one neuron and the response of the other. We also analyze how the synchronization depends on this delay. A rich spectrum of synchronized behaviors is possible for electrically coupled neurons and for inhibitory coupling between neurons. In synchronous neurons one typically sees chaotic motion of the coupled neurons. Excitatory coupling produces essentially periodic voltage trajectories, which are also synchronized. We display and discuss these synchronized behaviors using two distance measures of the synchronization.

On the origin of reproducible sequential activity in neural circuits

Robustness and reproducibility of sequential spatio-temporal responses is an essential feature of many neural circuits in sensory and motor systems of animals. The most common mathematical images of dynamical regimes in neural systems are fixed points, limit cycles, chaotic attractors, and continuous attractors (attractive manifolds of neutrally stable fixed points). These are not suitable for the description of reproducible transient sequential neural dynamics. In this paper we present the concept of a stable heteroclinic sequence (SHS), which is not an attractor. SHS opens the way for understanding and modeling of transient sequential activity in neural circuits. We show that this new mathematical object can be used to describe robust and reproducible sequential neural dynamics. Using the framework of a generalized high-dimensional Lotka-Volterra model, that describes the dynamics of firing rates in an inhibitory network, we present analytical results on the existence of the SHS in the phase space of the network. With the help of numerical simulations we confirm its robustness in presence of noise in spite of the transient nature of the corresponding trajectories. Finally, by referring to several recent neurobiological experiments, we discuss possible applications of this new concept to several problems in neuroscience.

Learning classification in the olfactory system of insects

We propose a theoretical framework for odor classification in the olfactory system of insects. The classification task is accomplished in two steps. The first is a transformation from the antennal lobe to the intrinsic Kenyon cells in the mushroom body. This transformation into a higher-dimensional space is an injective function and can be implemented without any type of learning at the synaptic connections. In the second step, the encoded odors in the intrinsic Kenyon cells are linearly classified in the mushroom body lobes. The neurons that perform this linear classification are equivalent to hyperplanes whose connections are tuned by local Hebbian learning and by competition due to mutual inhibition. We calculate the range of values of activity and size fo the network required to achieve efficient classification within this scheme in insect olfaction. We are able to demonstrate that biologically plausible control mechanisms can accomplish efficient classification of odors.

Dynamical model of long-term synaptic plasticity

Long-term synaptic plasticity leading to enhancement in synaptic efficacy (long-term potentiation, LTP) or decrease in synaptic efficacy (long-term depression, LTD) is widely regarded as underlying learning and memory in nervous systems. LTP and LTD at excitatory neuronal synapses are observed to be induced by precise timing of pre- and postsynaptic events. Modification of synaptic transmission in long-term plasticity is a complex process involving many pathways; for example, it is also known that both forms of synaptic plasticity can be induced by various time courses of Ca2+ introduction into the postsynaptic cell. We present a phenomenological description of a two-component process for synaptic plasticity. Our dynamical model reproduces the spike time-dependent plasticity of excitatory synapses as a function of relative timing between pre- and postsynaptic events, as observed in recent experiments. The model accounts for LTP and LTD when the postsynaptic cell is voltage clamped and depolarized (LTP) or hyperpolarized (LTD) and no postsynaptic action potentials are evoked. We are also able to connect our model with the Bienenstock, Cooper, and Munro rule. We give model predictions for changes in synaptic strength when periodic spike trains of varying frequency and Poisson distributed spike trains with varying average frequency are presented pre- and postsynaptically. When the frequency of spike presentation exceeds ≈30–40 Hz, only LTP is induced.

Extended dynamic clamp: controlling up to four neurons using a single desktop computer and interface

The dynamic clamp protocol allows an experimenter to simulate the presence of membrane conductances in, and synaptic connections between, biological neurons. Existing protocols and commercial ADC/DAC boards provide ready control in and between < or =2 neurons. Control at >2 sites is desirable when studying neural circuits with serial or ring connectivity. Here, we describe how to extend dynamic clamp control to four neurons and their associated synaptic interactions, using a single IBM-compatible PC, an ADC/DAC interface with two analog outputs, and an additional demultiplexing circuit. A specific C++ program, DYNCLAMP4, implements these procedures in a Windows environment, allowing one to change parameters while the dynamic clamp is running. Computational efficiency is increased by varying the duration of the input-output cycle. The program simulates < or =8 Hodgkin-Huxley-type conductances and < or =18 (chemical and/or electrical) synapses in < or =4 neurons and runs at a minimum update rate of 5 kHz on a 450 MHz CPU. (Increased speed is possible in a two-neuron version that does not need auxiliary circuitry). Using identified neurons of the crustacean stomatogastric ganglion, we illustrate on-line parameter modification and the construction of three-member synaptic rings.