For classical billiards, we suggest that a matrix of action or length of trajectories in conjunction with statistical measures, level spacing distribution and spectral rigidity, can be used to distinguish chaotic from integrable systems. As examples of 2D chaotic billiards, we considered the Bunimovich stadium billiard and the Sinai billiard. In the level spacing distribution and spectral rigidity, we found GOE behaviour consistent with predictions from random matrix theory. We studied transport properties and computed a diffusion coefficient. For the Sinai billiard, we found normal diffusion, while the stadium billiard showed anomalous diffusion behaviour. As example of a 2D integrable billiard, we considered the rectangular billiard. We found very rigid behaviour with strongly correlated spectra similar to a Dirac comb. These findings present numerical evidence for universality in level spacing fluctuations to hold in classically integrable systems and in classically fully chaotic systems.

Two main excitatory synapses are formed at the dendritic arbor of first-order nuclei thalamocortical (TC) neurons. Ascending sensory axons primarily establish contacts at large proximal dendrites, whereas descending corticothalamic fibers form synapses on thin distal dendrites. With the use of a multicomparment computational model based on fully reconstructed TC neurons from the ventroposterolateral nucleus of the cat, we compared local responses at the site of stimulation as well as somatic responses induced by both α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid receptor (AMPAR)- and N-methyl-D-aspartate receptor (NMDAR)-mediated currents. We found that AMPAR-mediated responses, when synapses were located at proximal dendrites, induced a larger depolarization at the level of soma, whereas NMDAR-mediated responses were more efficient for synapses located at distal dendrites. The voltage transfer and transfer impedance were higher for NMDAR than for AMPAR activation at any location. For both types of synaptic current and for both input locations at the dendritic arbor, somatic responses were characterized by a low variability despite the large variability found in local responses in dendrites. The large neurons had overall smaller somatic responses than small neurons, but this relation was not found in local dendritic responses. We conclude that in TC cells, the dendritic location of small synaptic inputs does not play a major role in the amplitude of a somatic response, but the size of the neuron does. The variability of response amplitude between cells was much larger than the variability within cells. This suggests possible functional segregation of TC neurons of different size.

We study chaotic behavior and diffusion in the 2D periodic Lorentz gas in the finite-horizon regime. The dynamical observable which we consider is the length of single particle's trajectories, which moves in a triangular array of rigid disks. To analyze the chaoticity of this system, we construct a matrix of the length of trajectories and perform a level spacing analysis of the spectrum of this matrix. We find that a universal behavior occurs both in level spacing distribution and spectral rigidity. In order to examine diffusion in this classical system, we investigate the variance of the length of trajectories versus number of collisions to disks. In the case where there is a finite-horizon, such a variance scales linearly with number of bounces. This shows that a normal diffusion exists and a central limit theorem is maintained in this regime.

We address an old problem in lattice gauge theory - the computation of the spectrum and wave functions of excited states. Our method is based on the Hamiltonian formulation of lattice gauge theory. As strategy, we propose to construct a stochastic basis of Bargmann link states, drawn from a physical probability density distribution. Then we compute transition amplitudes between stochastic basis states. From a matrix of transition elements we extract energy spectra and wave functions. We apply this method to U(1)$_{2+1}$ lattice gauge theory. We test the method by computing the energy spectrum, wave functions and thermodynamical functions of the electric Hamiltonian of this theory and compare them with analytical results. We observe a reasonable scaling of energies and wave functions in the variable of time. We also present first results on a small lattice for the full Hamiltonian including the magnetic term.

Chloride homeostasis is a critical determinant of the strength and robustness of inhibition mediated by GABA(A) receptors (GABA(A)Rs). The impact of changes in steady state Cl(-) gradient is relatively straightforward to understand, but how dynamic interplay between Cl(-) influx, diffusion, extrusion and interaction with other ion species affects synaptic signaling remains uncertain. Here we used electrodiffusion modeling to investigate the nonlinear interactions between these processes. Results demonstrate that diffusion is crucial for redistributing intracellular Cl(-) load on a fast time scale, whereas Cl(-)extrusion controls steady state levels. Interaction between diffusion and extrusion can result in a somato-dendritic Cl(-) gradient even when KCC2 is distributed uniformly across the cell. Reducing KCC2 activity led to decreased efficacy of GABA(A)R-mediated inhibition, but increasing GABA(A)R input failed to fully compensate for this form of disinhibition because of activity-dependent accumulation of Cl(-). Furthermore, if spiking persisted despite the presence of GABA(A)R input, Cl(-) accumulation became accelerated because of the large Cl(-) driving force that occurs during spikes. The resulting positive feedback loop caused catastrophic failure of inhibition. Simulations also revealed other feedback loops, such as competition between Cl(-) and pH regulation. Several model predictions were tested and confirmed by [Cl(-)](i) imaging experiments. Our study has thus uncovered how Cl(-) regulation depends on a multiplicity of dynamically interacting mechanisms. Furthermore, the model revealed that enhancing KCC2 activity beyond normal levels did not negatively impact firing frequency or cause overt extracellular K(-) accumulation, demonstrating that enhancing KCC2 activity is a valid strategy for therapeutic intervention.

Morphological features of the dendritic arborization can affect neuronal responses and thus the input-output function of a particular neuron. In this study, morphological data of eight fully reconstructed thalamocortical (TC) neurons from the ventroposterolateral (VPL) nucleus of adult cats have been analyzed. We examined several geometrical and topological parameters, which have been previously shown to have a high impact on the neuron firing pattern and propagation of signals in the dendritic tree. In addition to well-known morphological parameters such as number of dendritic trees (8.3 +/- 1.5) and number of branching points (80-120), we investigated the distribution of dendritic membrane area, branching points, geometrical ratio, asymmetry index, and mean path length for all subtrees of the TC neurons. We demonstrate that due to extensive branching in proximal and middle dendritic sections, the maximum value of the dendritic area distribution is reached at 120-160 mum from the soma. Our analysis reveals that TC neurons are highly branched cells and their dendritic branching pattern does not follow Rall's 3/2 power rule; average values at proximal vs. distal dendritic sections were different. We also found that the dendritic branching pattern of each subtree of the cell had a wide range in symmetry index, whereas the mean path length did not show a large variation through the dendritic arborizations.

Chemically mediated synaptic transmission results from fusion of synaptic vesicles with the presynaptic plasma membrane, subsequent release of the vesicular content into the cleft and binding to postsynaptic receptors. Previous modelling studies of excitatory neurotransmitter glutamate were based on simplified geometries failing to account for the biologically realistic synaptic environment, in particular, the presence of astrocytes, the geometry of extracellular space, and the neurotransmitter uptake mechanism. Using 3-dimensional reconstructions of hippocampal glutamatergic synapses including the surrounding astrocytic processes we have developed a biologically realistic model to analyse receptor activation in different conditions. We used the finite element method to simulate glutamate release, analyse glutamate diffusion following single and multiple vesicle release and binding at the postsynaptic site to AMPA and NMDA receptors. We demonstrate that: (1) the transmitter diffusion is highly temperature-sensitive; (2) release conditions and geometry more specifically affect AMPARs than NMDARs; (3) the sensitivities of AMPARs and NMDARs to simultaneous vesicular release are different; (4) in the case of multivesicle neurotransmitter release with variable delays, the binding of glutamate to AMPARs is additive up to 1 ms after the release, then becomes independent, but to NMDARs the binding is additive up to 33 ms; (5) the number of AMPARs varies more than the number of NMDRs in response to the input firing patterns; (6) the presence of astrocytes effectively blocks synaptic cross-talk; and (7) synaptic cross-talk, mediated by NMDARs but not AMPARs, is only possible after quasi-simultaneous multivesicular release at physiological temperature (35 degrees C) without intervening astrocytes, but not at 25 degrees C. Our simulations demonstrate the importance of temperature and ultrastructural synaptic environment in synaptic transmission and synaptic cross-talk.

We study chaotic behavior of a classical optical stadium billiard model. We construct a matrix of time-of-travel along trajectories corresponding to a set of boundary points. We carry out a level spacing fluctuation analysis and compute the Dyson–Mehta spectral rigidity. The distribution of time-of-travel is approximately described by a Gaussian. The results for level spacing distribution and spectral rigidity show universal behavior.

Cognitive processes in the brain, like learning, formation of memory, recovery of memorized images, classification of objects have two features: First, there is no supervisor in the brain who controls these processes. Second there is a hugh number of neurons (10^{6} to 10^{10}) involved in those cognitive tasks. For this reason, the search of understanding cognitive processes uses models built from a large number of neurons, but very much simplified neurons. The so-called neural networks have been quite successful in describing certain aspects of brain functions, like the mechanism of associative memory or recently the prediction of epileptic seizures. At hand of the Kohonen network we discuss the treatment of information in the brain, in particular how the brain organizes such information without supervisor. Recently, networks of small-world and scale-free architecture came into focus. There is evidence indicating that the brain (cat cortex, macaque cortex, human brain) uses such connectivity architecture. Tasks like treatment of information, learning and classification take advantage of such scale-free and small-world connectivity and thus play a potentially important role in self-organization of the brain.

We suggest a new method to compute the spectrum and wave functions of excited states.We construct a stochastic basis of Bargmann link states, drawn from a physical probability density distribution and compute transition amplitudes between stochastic basis states.From such transition matrix we extract wave functions and the energy spectrum.We apply this method to U(1) 2+1 lattice gauge theory.As a test we compute the energy spectrum, wave functions and thermodynamical functions of the electric Hamiltonian and compare it with analytical results.We find excellent agreement.We observe scaling of energies and wave functions in the variable of time.We also present first results on a small lattice for the full Hamiltonian including the magnetic term.

For classical billiards we suggest that a matrix of action or length of trajectories in conjunction with statistical measures, level spacing distribution and spectral rigidity, can be used to distinguish chaotic from integrable systems. As examples of 2D chaotic billiards we considered the Bunimovich stadium billiard and the Sinai billiard. In the level spacing distribution and spectral rigidity we found GOE behaviour consistent with predictions from random matrix theory. We studied transport properties and computed a diffusion coefficient. For the Sinai billiard, we found normal diffusion, while the stadium billiard showed anomalous diffusion behaviour. As example of a 2D integrable billiard we considered the rectangular billiard. We found very rigid behaviour with strongly correlated spectra similar to a Dirac comb. These findings present numerical evidence for universality in level spacing fluctuations to hold in classically integrable systems and in classically fully chaotic systems.

We use the quantum action to study quantum chaos at finite temperature. We present a numerical study of a classically chaotic 2-D Hamiltonian system harmonic oscillators with anharmonic coupling. We construct the quantum action non-perturbatively and find temperature dependent quantum corrections in the action parameters. We compare Poincaré sections of the quantum action at finite temperature with those of the classical action. ∗Corresponding author, Email: hkroger@phy.ulaval.ca

The influence of calcium channel distribution and geometry of the thalamocortical cell upon its tonic firing and the low threshold spike (LTS) generation was studied in a 3-compartment model, which represents soma, proximal and distal dendrites as well as in multi-compartment model using the morphology of a real reconstructed neuron. Using an uniform distribution of Ca2+ channels, we determined the minimal number of low threshold voltage-activated calcium channels and their permeability required for the onset of LTS in response to a hyperpolarizing current pulse. In the 3-compartment model, we found that the channel distribution influences the firing pattern only in the range of 3% below the threshold value of total T-channel density. In the multi-compartmental model, the LTS could be generated by only 64% of unequally distributed T-channels compared to the minimal number of equally distributed T-channels. For a given channel density and injected current, the tonic firing frequency was found to be inversely proportional to the size of the cell. However, when the Ca2+ channel density was elevated in soma or proximal dendrites, then the amplitude of LTS response and burst spike frequencies were determined by the ratio of total to threshold number of T-channels in the cell for a specific geometry.

We propose a variant of the Kohonen model to study principles of self-organization in a model of young brain. We suggest a function to measure aquired knowledge and use it to auto-adapt the topology of neuronal connectivity, yielding substantial organizational improvement relative to the standard model. In the early phase of organization with most intense learning, we observe that neural connectivity is of Small World type, which is very efficient to organize neurons in response to stimuli. In analogy to human brain where pruning of neural connectivity occurs in early life, this feature is found to stabilize neuronal response to stimuli.

The low-threshold calcium current (IT) underlies burst generation in thalamocortical (TC) relay cells and plays a central role in the genesis of synchronized oscillations by thalamic circuits. Ascending and descending inputs to thalamic relay cells arrive on the dendritic tree, thus the study of synaptic integration in model cells requires simulations incorporating the electrically active properties of the dendritic tree. We developed a 3-compartment model of a thalamocortical cell to consider effects of dendritic currents on the response of the cell. First, we attribute uniform T-channel distribution for all compartments in the model, then we find a threshold value of channel permeability, Pca = 1.56e-4 (cm/sec) that enables generation of a low-threshold calcium spike (LTS). By multiplying the permeability of each section by its area we estimated the threshold number of channels necessary to reproduce an LTS. While we kept the total number of channels constant, we attributed different calcium permeability to the different compartments. Our simulations show, independent of the Ca2+ channel distribution, for a total channel number below the threshold value that the cell gives a passive response and above the threshold, the model reproduces the LTS spike. In a small range below the threshold the difference between uniform and non uniform distributions becomes visible. In the range of 1–2% below threshold a uniform distribution of T-channels produces a passive response while a non-uniform distribution reproduces an LTS response. A comparison with experimental data of firing patterns and the I-V curve of the T-current shows a non-uniform distribution with higher density in sections near the soma better represents the experimental data. However, the geometry of the I-V-curve of the T-current strongly depends on the quality of the voltage clamp. Depending on the electrode resistance, the maximum of the I-V-curve can change from (6 nA, -60 mV) to (3 nA, -40 mV). In addition we investigated the influence of the cell size on firing patterns. The model shows that with the same number of channels, the frequency of tonic firing increases with a decrease in the cell size, while the LTS response increases with an increase in cell size.

Local field potentials (LFPs) are routinely measured experimentally in brain tissue, and exhibit strong low-pass frequency filtering properties, with high frequencies (such as action potentials) being visible only at very short distances (approximately 10 microm) from the recording electrode. Understanding this filtering is crucial to relate LFP signals with neuronal activity, but not much is known about the exact mechanisms underlying this low-pass filtering. In this paper, we investigate a possible biophysical mechanism for the low-pass filtering properties of LFPs. We investigate the propagation of electric fields and its frequency dependence close to the current source, i.e., at length scales in the order of average interneuronal distances. We take into account the presence of a high density of cellular membranes around current sources, such as glial cells. By considering them as passive cells, we show that under the influence of the electric source field, they respond by polarization. Because of the finite velocity of ionic charge movements, this polarization will not be instantaneous. Consequently, the induced electric field will be frequency-dependent, and much reduced for high frequencies. Our model establishes that this situation is analogous to an equivalent RC circuit, or better yet a system of coupled RC circuits. We present a number of numerical simulations of an induced electric field for biologically realistic values of parameters, and show the frequency filtering effect as well as the attenuation of extracellular potentials with distance. We suggest that induced electric fields in passive cells surrounding neurons are the physical origin of frequency filtering properties of LFPs. Experimentally testable predictions are provided allowing us to verify the validity of this model.

We investigate the dynamics of gravity coupled to a scalar field using a non-canonical form of the kinetic term. It is shown that its singular point represents an attractor for classical solutions and the stationary value of the field may occur distant from the minimum of the potential. In this paper properties of universes with such stationary states are considered. We reveal that such state can be responsible for modern dark energy density.

We investigate the bound states of the Yukawa potential V (r)=−λexp(−αr)/r, using different algorithms: solving the Schrödinger equation numerically and our Monte Carlo Hamiltonian approach. There is a critical α = αC, above which no bound state exists. We study the relation between αC and λ for various angular momentum quantum number l, and find in atomic units, αC(l) = λ[A1 exp(−l/B1) + A2 exp(−l/B2)], with A1 = 1.020(18), B1 = 0.443(14), A2 = 0.170(17), and B2 = 2.490(180).

We discuss the questions: How to compare quantitatively classical chaos with quantum chaos? Which one is stronger? What are the underlying physical reasons?