Dmitry Tsigankov

A unifying model for activity-dependent and activity-independent mechanisms predicts complete structure of topographic maps in ephrin-A deficient mice

Axons of retinal ganglion cells establish orderly projections to the superior colliculus of the midbrain. Axons of neighboring cells terminate proximally in the superior colliculus thus forming a topographically precise representation of the visual world. Coordinate axes are encoded in retina and in the target through graded expression of chemical labels. Additional sharpening of projections is provided by electric activity, which is correlated between neighboring axons. Here we propose a quantitative model, which allows combining the effects of chemical labels and correlated activity in a single approach. Using this model we study a complete structure of two-dimensional topographic maps in mutant mice, in which the label encoding the horizontal retinal coordinate ephrin-A is reduced/eliminated. We show that topographic maps in ephrin-A deficient mice display a granular structure, with the regions of smooth mapping separated by linear discontinuities reminiscent of fractures observed in the maps of preferred orientation.

A stochastic model for retinocollicular map development.

BackgroundWe examine results of gain-of-function experiments on retinocollicular maps in knock-in mice [Brown et al. (2000) Cell 102:77]. In wild-type mice the temporal-nasal axis of retina is mapped to the rostral-caudal axis of superior colliculus. The established map is single-valued, which implies that each point in retina maps to a unique termination zone in superior colliculus. In homozygous Isl2/EphA3 knock-in mice the map is double-valued, which means that each point on retina maps to two termination zones in superior colliculus. This is because about 50 percent of cells in retina express Isl2, and two types of projections, wild-type and Isl2/EphA3 positive, form two branches of the map. In heterozygous Isl2/EphA3 knock-ins the map is intermediate between the homozygous and wild-type: it is single-valued in temporal and double-valued in the nasal parts of retina. In this study we address possible reasons for such a bifurcation of the map.ResultsWe study the map formation using stochastic model based on Markov chains. In our model the map undergoes a series of reconstructions with probabilities dependent upon a set of chemical cues. Our model suggests that the map in heterozygotes is single-valued in temporal region of retina for two reasons. First, the inhomogeneous gradient of endogenous receptor in retina makes the impact of exogenous receptor less significant in temporal retina. Second, the gradient of ephrin in the corresponding region of superior colliculus is smaller, which reduces the chemical signal-to-noise ratio. We predict that if gradient of ephrin is reduced by a genetic manipulation, the single-valued region of the map should extend to a larger portion of temporal retina, i.e. the point of transition between single-and doulble-valued maps should move to a more nasal position in Isl2-EphA3 heterozygotes.ConclusionsWe present a theoretical model for retinocollicular map development, which can account for intriguing behaviors observed in gain-of-function experiments by Brown et al., including bifurcation in heterozygous Isl2/EphA3 knock-ins. The model is based on known chemical labels, axonal repulsion/competition, and stochasticity. Possible mapping in Isl2/EphB knock-ins is also discussed.

Optimal axonal and dendritic branching strategies during the development of neural circuitry.

In developing brain, axons and dendrites are capable of connecting to each other with high precision. Imaging of axonal and dendritic dynamics in vivo shows that the majority of axonal and dendritic branches are formed ‘in error’, only to be retracted later. The functional significance of the overproduction of branches is not clear. Here we show that branching of both axons and dendrites can accelerate finding appropriate synaptic targets during the development of neuronal circuitry. We suggest that branching rules implemented by axons and dendrites minimize the number of erroneous branches. We find that optimal branching rules are different for axons and dendrites in agreement with experimentally observed branch dynamics. Thus, our studies suggest that the developing neural system employs a set of sophisticated computational strategies that facilitate the formation of required circuitry in the fastest and most frugal way.

How to find decision makers in neural networks

Nervous systems often face the problem of classifying stimuli and making decisions based on these classifications. The neurons involved in these tasks can be characterized as sensory or motor, according to their correlation with sensory stimulus or motor response. In this study we define a third class of neurons responsible for making perceptual decisions. Our mathematical formalism enables the weighting of neuronal units according to their contribution to decision making, thus narrowing the field for more detailed studies of underlying mechanisms. We develop two definitions of a contribution to decision making. The first definition states that decision making activity can be found at the points of emergence for behavioral correlations in the system. The second definition involves the study of propagation of noise in the network. The latter definition is shown to be equivalent to the first one in the cases when they can be compared. Our results suggest a new approach to analyzing decision making networks.

The molecular basis for the development of neural maps

Neural development leads to the establishment of precise connectivity in the nervous system. By contrasting the information capacities of cortical connectivity and the genome, we suggest that simplifying rules are necessary in order to create cortical connections from the limited set of instructions contained in the genome. One of these rules may be employed by the visual system, where connections are formed on the basis of the interplay of molecular gradients and activity‐dependent synaptic plasticity. We show how a simple model that accounts for such interplay can create both neural topographic maps and more complex patterns of ocular dominance, that is, the segregated binary mixture of projections from two eyes converging in the same visual area. With regard to the ocular dominance patterns, we show that pattern orientation may be instructed by the direction of the gradients of molecular labels. We also show that the periodicity of ocular dominance patterns may result from the interplay of the effects of molecular gradients and correlated neural activity. Overall, we propose that simple mechanisms can account for the formation of apparently complex features of neuronal connections.

A stochastic model describing transport of PSD-95 molecules in spiny dendrites provides the basis for synaptic plasticity.

Molecules in neurons are in a state far from equilibrium. Therefore their transport properties are strongly affected by fluctuations. We present a model of stochastic molecular transport in neurons which have their synapses located in the spines of a dendrite. In this model we assume that the molecules perform a random walk between the spines that trap the walkers. If the molecules are assumed to interact with each other inside the spines, the trapping time in each spine depends on the number of molecules in the respective trap. The corresponding mathematical problem has non-trivial solutions even in the absence of external disorder due to self-organization phenomena. We obtain the stationary distributions of the number of walkers in the traps for different kinds of on-site interactions between the walkers and furthermore analyze how birth and death processes of the random walkers affect these distributions. We apply this model to describe the dynamics of the PSD-95 proteins in spiny dendrites. PSD-95 is the most abundant molecule in the post-synaptic density (PSD) located in the spines. It is observed that these molecules have high turnover rates and that neighboring spines are constantly exchanging individual molecules. Thus we predict the distribution of PSD-95 cluster sizes that determine the size of the synapse and thus the synaptic strength. Finally, we show that in the model non-equilibrium inter-spine dynamics of PSD-95 molecules can provide the basis for locally controlled synaptic plasticity through activity-dependent ubiquitinization of PSD-95 molecules.

Model of inter-spine dynamics of PSD-95 molecules and its application to synaptic plasticity

We present a model of stochastic molecular transport in spiny dendrites. In this model the molecules perform a random walk between the spines that trap the walkers. If the molecules interact with each other inside the spines the trapping time in each spine depends on the number of molecules in the respective trap. The corresponding mathematical problem has non-trivial solutions even in the absence of external disorder due to self-organization phenomenon. We obtain the stationary distributions of the number of walkers in the traps for different kinds of on-site interactions between the walkers. We analyze how birth and death processes of the random walkers affect these distributions. We apply this model to describe the dynamics of the PSD-95 proteins in spiny dendrites. PSD-95 is the most abundant molecule in the post-synaptic density (PSD) located in the spines. It is observed that these molecules have high turnover rates and that neighboring spines are constantly exchanging individual molecules. We propose that the geometry of individual PSD-95 clusters determines the dependence of trapping times on the number of molecules inside the trap and thus can vary from spine to spine. Furthermore, we suggest that activity-dependent reorganization of the PSD-95 cluster can lead to synaptic plasticity in a form of long-term potentiation (LTP). In the model this is achieved by spine specific activity-dependent ubiquitinization of PSD-95 molecules, which transiently reduces the amount of PSD-95 in the spine but change the geometry of the PSD-95 cluster in such a way that self-organization process results in the overall increase of the number of PSD-95 molecules associated with LTP. We also show that such a dynamics of the PSD-95 molecules can set up the conditions for anomalous diffusive transport inside spiny dendrites and predict the distribution of the PSD sizes which has features of both exponential and Poisson distributions.

Is the formation of ocular dominance patterns instructed by molecular labels

The formation of ocular dominance patterns in visual cortex is thought to be driven by electrical activity of the projecting LGN neurons. Indeed, theoretical modeling has shown that lateral cortical activity-dependent interactions are capable of producing such a periodic pattern of regions dominated by inputs from left or right eye [1]. In this view, the ocular dominance emerges from the interplay of excitatory and inhibitory connections. The major characteristic of these structures, the width of the ocular dominance regions, is then determined by the spatial profile of the lateral interactions. Experimental attempts to confirm this view by altering the strength of the inhibitory connections have shown [2] that observed changes of the ocular dominance width are inconsistent with the models central predictions. When the strength of inhibitory connections is increased and thus the characteristic length scale of the lateral interaction profile is decreased, the width of the ocular dominance was found to increase, indicating the opposite behavior to one predicted by the model. Here we resolve this paradox by investigating how other factors, such as interactions between chemical labels, can participate in the formation of ocular dominance patterns.