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A recent theoretical emphasis on complex interactions within neural systems underlying consciousness has been accompanied by proposals for the quantitative characterization of these interactions. In this article, we distinguish key aspects of consciousness that are amenable to quantitative measurement from those that are not. We carry out a formal analysis of the strengths and limitations of three quantitative measures of dynamical complexity in the neural systems underlying consciousness: neural complexity, information integration, and causal density. We find that no single measure fully captures the multidimensional complexity of these systems, and all of these measures have practical limitations. Our analysis suggests guidelines for the specification of alternative measures which, in combination, may improve the quantitative characterization of conscious neural systems. Given that some aspects of consciousness are likely to resist quantification altogether, we conclude that a satisfactory theory is likely to be one that combines both qualitative and quantitative elements.

The functional and biophysical properties of a persistent sodium current (I(NaP)) previously proposed to participate in the generation of subthreshold oscillations and burst discharge in mesencephalic trigeminal sensory neurons (Mes V) were investigated in brain stem slices (rats, p7-p12) using whole cell patch-clamp methods. I(NaP) activated around -76 mV and peaked at -48 mV, with V1/2 of -58.7 mV. Ramp voltage-clamp protocols showed that I(NaP) undergoes time- as well as voltage-dependent inactivation and recovery from inactivation in the range of several seconds (tau(onset) = 2.04 s, tau(recov) = 2.21 s). Riluzole (< or =5 microM) substantially reduced I(NaP), membrane resonance, postinhibitory rebound (PIR), and subthreshold oscillations, and completely blocked bursting, but produced modest effects on the fast transient Na+ current (I(NaT)). Before complete cessation, burst cycle duration was increased substantially, while modest and inconsistent changes in burst duration were observed. The properties of the I(NaT) were obtained and revealed that the amplitude and voltage dependence of the resulting "window current" were not consistent with those of the observed I(NaP) recorded in the same neurons. This suggests an additional mechanism for the origin of I(NaP). A neuronal model was constructed using Hodgkin-Huxley parameters obtained experimentally for Na+ and K+ currents that simulated the experimentally observed membrane resonance, subthreshold oscillations, bursting, and PIR. Alterations in the model g(NaP) parameters indicate that I(NaP) is critical for control of subthreshold and suprathreshold Mes V neuron membrane excitability and burst generation.

A model is presented that reproduces spiking and bursting behavior of known types of cortical neurons. The model combines the biologically plausibility of Hodgkin-Huxley-type dynamics and the computational efficiency of integrate-and-fire neurons. Using this model, one can simulate tens of thousands of spiking cortical neurons in real time (1 ms resolution) using a desktop PC.

We propose a novel architecture of an oscillatory neural network that consists of phase-locked loop (PLL) circuits. It stores and retrieves complex oscillatory patterns as synchronized states with appropriate phase relations between neurons.

Many scientists believe that all pulse-coupled neural networks are toy models that are far away from the biological reality. We show here, however, that a huge class of biophysically detailed and biologically plausible neural-network models can be transformed into a canonical pulse-coupled form by a piece-wise continuous, possibly noninvertible, change of variables. Such transformations exist when a network satisfies a number of conditions; e.g., it is weakly connected; the neurons are Class 1 excitable (i.e., they can generate action potentials with an arbitrary small frequency); and the synapses between neurons are conventional (i.e., axo-dendritic and axo-somatic). Thus, the difference between studying the pulse-coupled model and Hodgkin-Huxley-type neural networks is just a matter of a coordinate change. Therefore, any piece of information about the pulse-coupled model is valuable since it tells something about all weakly connected networks of Class 1 neurons. For example, we show that the pulse-coupled network of identical neurons does not synchronize in-phase. This confirms Ermentrout's result that weakly connected Class 1 neurons are difficult to synchronize, regardless of the equations that describe dynamics of each cell.

We study pulse-coupled neural networks that satisfy only two assumptions: each isolated neuron fires periodically, and the neurons are weakly connected. Each such network can be transformed by a piece-wise continuous change of variables into a phase model, whose synchronization behavior and oscillatory associative properties are easier to analyze and understand. Using the phase model, we can predict whether a given pulse-coupled network has oscillatory associative memory, or what minimal adjustments should be made so that it can acquire memory. In the search for such minimal adjustments we obtain a large class of simple pulse-coupled neural networks that can memorize and reproduce synchronized temporal patterns the same way a Hopfield network does with static patterns. The learning occurs via modification of synaptic weights and/or synaptic transmission delays.

This is the second of two articles devoted to analyzing the relationship between synaptic organizations (anatomy) and dynamical properties (function) of networks of neural oscillators near multiple supercritical Andronov-Hopf bifurcation points. Here we analyze learning processes in such networks. Regarding learning dynamics, we assume (1) learning is local (i.e. synaptic modification depends on pre- and postsynaptic neurons but not on others), (2) synapses modify slowly relative to characteristic neuron response times, (3) in the absence of either pre- or postsynaptic activity, the synapse weakens (forgets). Our major goal is to analyze all synaptic organizations of oscillatory neural networks that can memorize and retrieve phase information or time delays. We show that such network have the following attributes: (1) the rate of synaptic plasticity connected with learning is determined locally by the presynaptic neurons, (2) the excitatory neurons must be long-axon relay neurons capable of forming distant connections with other excitatory and inhibitory neurons, (3) if inhibitory neurons have long axons, then the network can learn, passively forget and actively unlearn information by adjusting synaptic plasticity rates.

We study weakly connected networks of neural oscillators near multiple Andronov-Hopf bifurcation points. We analyze relationships between synaptic organizations (anatomy) of the networks and their dynamical properties (function). Our principal assumptions are: (1) Each neural oscillator comprises two populations of neurons; excitatory and inhibitory ones; (2) activity of each population of neurons is described by a scalar (one-dimensional) variable; (3) each neural oscillator is near a nondegenerate supercritical Andronov-Hopf bifurcation point; (4) the synaptic connections between the neural oscillators are weak. All neural networks satisfying these hypotheses are governed by the same dynamical system, which we call the canonical model. Studying the canonical model shows that: (1) A neural oscillator can communicate only with those oscillators which have roughly the same natural frequency. That is, synaptic connections between a pair of oscillators having different natural frequencies are functionally insignificant. (2) Two neural oscillators having the same natural frequencies might not communicate if the connections between them are from among a class of pathological synaptic configurations. In both cases the anatomical presence of synaptic connections between neural oscillators does not necessarily guarantee that the connections are functionally significant. (3) There can be substantial phase differences (time delays) between the neural oscillators, which result from the synaptic organization of the network, not from the transmission delays. Using the canonical model we can illustrate self-ignition and autonomous quiescence (oscillator death) phenomena. That is, a network of passive elements can exhibit active properties and vice versa. We also study how Dale's principle affects dynamics of the networks, in particular, the phase differences that the network can reproduce. We present a complete classification of all possible synaptic organizations from this point of view. The theory developed here casts some light on relations between synaptic organization and functional properties of oscillatory networks. The major advantage of our approach is that we obtain results about all networks of neural oscillators, including the real brain. The major drawback is that our findings are valid only when the brain operates near a critical regime, viz. for a multiple Andronov-Hopf bifurcation.

We consider a model of a neural network where the individual cells interact only by releasing and absorbing the molecules of a neuromediator. We show that such a system can realize the function of associative memory. A learning mechanism based on the chemotaxis is proposed and numerically investigated.