Instantons in the working memory: implications for schizophrenia

The influence of the synaptic channel properties on the stability of delayed activity maintained by recurrent neural network is studied. The duration of excitatory post-synaptic current (EPSC) is shown to be essential for the global stability of the delayed response. NMDA receptor channel is a much more reliable mediator of the reverberating activity than AMPA receptor, due to a longer EPSC. This allows to interpret the deterioration of working memory observed in the NMDA channel blockade experiments. The key mechanism leading to the decay of the delayed activity originates in the unreliability of the synaptic transmission. The optimum fluctuation of the synaptic conductances leading to the decay is identified. The decay time is calculated analytically and the result is confirmed computationally

On the scaling law for cortical magnification factor

Primate visual system samples different parts of the world unevenly. The part of the visual scene corresponding to the eye center is represented densely, while away from the center the sampling becomes progressively sparser. Such distribution allows a more effective use of the limited transfer rate of the optic nerve, since animals can aim area centralis (AC) at the relevant position in the scene by performing saccadic eye movements. To locate a new saccade target the animal has to sample the corresponding region of the visual scene, away from AC. In this work we derive the sampling density away from AC, which optimizes the trajectory of saccadic eye movements. We obtain the scaling law for the sampling density as a function of eccentricity, which results from the evolutionary pressure to locate the target in the shortest time under the constraint of limited transfer rate of the optic nerve. In case of very small AC the visual scene is optimally represented by logarithmic conformal mapping, in which geometrically similar circular bands around AC are equally represented by the visual system. We also obtain corrections to the logarithmic scaling for the case of a larger AC and compare them to experimental findings

Sperry versus Hebb: Topographic mapping in Isl2/EphA3 mutant mice

In wild-type mice axons of retinal ganglion cells establish topographically precise projection to the superior colliculus of the midbrain. This implies that axons of neighboring retinal ganglion cells project to the proximal locations in the target. The precision of topographic projection is a result of combined effects of molecular labels, such as Eph receptors and ephrins, and correlated electric activity. In the Isl2/EphA3 mutant mice the expression levels of molecular labels is changed. As a result the topographic projection is rewired so that the neighborhood relationships between retinal cell axons are disrupted. Here we argue that the effects of correlated activity presenting themselves in the form of Hebbian learning rules can facilitate the restoration of the topographic connectivity even when the molecular labels carry conflicting instructions. This occurs because the correlations in electric activity carry information about retinal cells' spatial location that is independent on molecular labels. We argue therefore that experiments in Isl2/EphA3 knock-in mice directly test the interaction between effects of molecular labels and correlated activity during the development of neural connectivity.Comment: 13 pages, 6 figure

An Exactly Solvable Model of Random Site-Specific Recombinations

Cre-lox and other systems are used as genetic tools to control site-specific recombination (SSR) events in genomic DNA. If multiple recombination sites are organized in a compact cluster within the same genome, a series of random recombination events may generate substantial cell specific genomic diversity. This diversity is used, for example, to distinguish neurons in the brain of the same multicellular mosaic organism, within the brainbow approach to neuronal connectome. In this paper we study an exactly solvable statistical model for SSR operating on a cluster of recombination sites. We consider two types of recombination events: inversions and excisions. Both of these events are available in the Cre-lox system. We derive three properties of the sequences generated by multiple recombination events. First, we describe the set of sequences that can in principle be generated by multiple inversions operating on the given initial sequence. We call this description the ergodicity theorem. On the basis of this description we calculate the number of sequences that can be generated from an initial sequence. This number of sequences is experimentally testable. Second, we demonstrate that after a large number of random inversions every sequence that can be generated is generated with equal probability. Lastly, we derive the equations for the probability to find a sequence as a function of time in the limit when excisions are much less frequent than inversions, such as in shufflon sequences

Neural integrator - a sandpile model

We investigated a model for the neural integrator based on hysteretic units connected by positive feedback. Hysteresis is assumed to emerge from the intrinsic properties of the cells. We consider the recurrent networks containing either bistable or multistable neurons. We apply our analysis to the oculomotor velocity-to-position neural integrator that calculates the eye positions from the inputs that carry information about eye angular velocity. Using the analysis of the system in the parameter space we show the following. The direction of hysteresis in the neuronal response may be reversed for the system with recurrent connections compared to the case of unconnected neurons. Thus, for the NMDA receptor based bistability the firing rates after ON saccades may be higher than after OFF saccades for the same eye position. We suggest that this is an emergent property due to the presence of global recurrent feedback. The reversal of hysteresis occurs only when the size of hysteresis differs from neuron to neuron. We also relate the macroscopic leak time-constant of the integrator to the rate of microscopic spontaneous noise-driven transitions in the hysteretic units. Finally, we argue that the presence of neurons with small hysteresis may remove the threshold for integration

Sparse incomplete representations: A novel role for olfactory granule cells

Mitral cells of the olfactory bulb form sparse representations of the odorants and transmit this information to the cortex. The olfactory code carried by the mitral cells is sparser than the inputs that they receive. In this study we analyze the mechanisms and functional significance of sparse olfactory codes. We consider a model of olfactory bulb containing populations of excitatory mitral and inhibitory granule cells. We argue that sparse codes may emerge as a result of self organization in the network leading to the precise balance between mitral cells' excitatory inputs and inhibition provided by the granule cells. We propose a novel role for the olfactory granule cells. We show that these cells can build representations of odorant stimuli that are not fully accurate. Due to the incompleteness in the granule cell representation, the exact excitation-inhibition balance is established only for some mitral cells leading to sparse responses of the mitral cell. Our model suggests a functional significance to the dendrodendritic synapses that mediate interactions between mitral and granule cells. The model accounts for the sparse olfactory code in the steady state and predicts that transient dynamics may be less sparse

Vortex Density of States and Absorption in Clean Layered Superconductors

We study the spectrum of the states localized in the vortex cores in the mixed state of clean layered superconductors. S-wave coupling is assumed. It is found that in a large region of parameters adjacent to the superclean case a new universal (i.e. independent of the density of impurities) class of level statistics arises. It is the circular unitary random matrix ensemble. The density of states for such conditions is calculated. The absorption resulting from the Landau-Zener transitions between these levels is different from the classical result for an isotropic three-dimensional system.Comment: 4 pages, 2 beautiful Postscript figures, submitted to PR

The Absence of the Fractional Quantum Hall Effect at High Landau Levels

We compare the energies of the Laughlin liquid and a charge density wave in a weak magnetic field for the upper Landau level filling factors $\nu_N = 1/3$ and $1/5$. The charge density wave period has been optimized and was found to be $\simeq 3R_c$, where $R_c$ is the cyclotron radius. We conclude that the optimal charge density wave is more energetically preferable than the Laughlin liquid for the Landau level numbers $N \ge 2$ at $\nu_N = 1/3$ and for $N \ge 3$ at $\nu_N = 1/5$. This implies that the $1/3$ fractional quantum Hall effect cannot be observed for $N \ge 2$, in agreement with the experiment.Comment: 12 pages, revtex, 2 PostScript figures are applied. Revised and corrected version. Also available at http://www.mnhep.umn.edu/~mfogler

Can repulsion be induced by attraction: a role of ephrin-B1 in retinotectal mapping?

We study a role of EphB receptors and their ligand ephrin-B1 in dorsal-ventral retinotopic mapping. Earlier studies suggested that ephrin-B1 acts as an attractant for EphB expressing axons. We address the results of the recent experiment in chick tectum (McLaughlin et al., 2003b) in which axons of retinal ganglion cells were shown to be repelled by high ephrin-B1 density. Thus it was proposed that ephrin-B1 might act as both attractant and repellent. We show that the same axonal behavior may follow from attraction to ephrin-B1 density and axonal competition for space. Therefore, we show how apparent repulsive interaction can be induced by a combination of attraction to the target and competitive interactions between axons. We suggest an experimental test that may distinguish repulsive interaction with the target from repulsion induced by attraction and competition.Comment: 4 page

Ocular dominance patterns and the wire length minimization: a numerical study

We study a mathematical model for ocular dominance patterns (ODPs) in primary visual cortex. This model is based on the premise that ODP is an adaptation to minimize the length of intra-cortical wiring. Thus we attempt to understand the existing ODPs by solving a wire length minimization problem. We divide all the neurons into two classes: left- and right-eye dominated. We find that segregation of neurons into monocular regions reduces wire length if the number of connections to the neurons of the same class (intraocular) differs from the number of interocular connections. The shape of the regions depends on the relative fraction of neurons in the two classes. We find that if both classes are almost equally represented, the optimal ODP consists of interdigitating stripes. If one class is less numerous than the other, the optimal ODP consists of patches of the less abundant class surrounded by the neurons of the other class. We predict that the transition from stripes to patches occurs when the fraction of neurons dominated by the underrepresented eye is about 40%. This prediction agrees with the data in macaque and Cebus monkeys. We also study the dependence of the periodicity of ODP on the parameters of our model