Moran Model of Spatial Alignment in Microbial Colonies.

We describe a spatial Moran model that captures mechanical interactions and directional growth in spatially extended populations. The model is analytically tractable and completely solvable under a mean-field approximation and can elucidate the mechanisms that drive the formation of population-level patterns. As an example we model a population of E. coli growing in a rectangular microfluidic trap. We show that spatial patterns can arise as a result of a tug-of-war between boundary effects and growth rate modulations due to cell-cell interactions: Cells align parallel to the long side of the trap when boundary effects dominate. However, when cell-cell interactions exceed a critical value, cells align orthogonally to the trap's long side. This modeling approach and analysis can be extended to directionally-growing cells in a variety of domains to provide insight into how local and global interactions shape collective behavior.

Democratization in a passive dendritic tree: an analytical investigation.

One way to achieve amplification of distal synaptic inputs on a dendritic tree is to scale the amplitude and/or duration of the synaptic conductance with its distance from the soma. This is an example of what is often referred to as "dendritic democracy". Although well studied experimentally, to date this phenomenon has not been thoroughly explored from a mathematical perspective. In this paper we adopt a passive model of a dendritic tree with distributed excitatory synaptic conductances and analyze a number of key measures of democracy. In particular, via moment methods we derive laws for the transport, from synapse to soma, of strength, characteristic time, and dispersion. These laws lead immediately to synaptic scalings that overcome attenuation with distance. We follow this with a Neumann approximation of Green's representation that readily produces the synaptic scaling that democratizes the peak somatic voltage response. Results are obtained for both idealized geometries and for the more realistic geometry of a rat CA1 pyramidal cell. For each measure of democratization we produce and contrast the synaptic scaling associated with treating the synapse as either a conductance change or a current injection. We find that our respective scalings agree up to a critical distance from the soma and we reveal how this critical distance decreases with decreasing branch radius.

Learning by structural remodeling in a class of single cell models.

Changes in neural connectivity are thought to underlie the most permanent forms of memory in the brain. We consider two models, derived from the clusteron (Mel, Adv Neural Inf Process Syst 4:35-42, 1992), to study this method of learning. The models show a direct relationship between the speed of memory acquisition and the probability of forming appropriate synaptic connections. Moreover, the strength of learned associations grows with the number of fibers that have taken part in the learning process. We provide simple and intuitive explanations of these two results by analyzing the distribution of synaptic activations. The obtained insights are then used to extend the model to perform novel tasks: feature detection, and learning spatio-temporal patterns. We also provide an analytically tractable approximation to the model to put these observations on a firm basis. The behavior of both the numerical and analytical models correlate well with experimental results of learning tasks which are thought to require a reorganization of neuronal networks.

Phase synchronization of chaotic systems with small phase diffusion.

The geometric theory of phase locking between periodic oscillators is extended to phase coherent chaotic systems. This approach explains the qualitative features of phase locked chaotic systems and provides an analytical tool for a quantitative description of the phase locked states. Moreover, this geometric viewpoint allows us to identify obstructions to phase locking even in systems with negligible phase diffusion, and to provide sufficient conditions for phase locking to occur. We apply these techniques to the Rössler system and a phase coherent electronic circuit and find that numerical results and experiments agree well with theoretical predictions.

Irrational phase synchronization.

We study the occurrence of physically observable phase locked states between chaotic oscillators and rotors in which the frequencies of the coupled systems are irrationally related. For two chaotic oscillators, the phenomenon occurs as a result of a coupling term which breaks the 2 pi invariance in the phase equations. In the case of rotors, a coupling term in the angular velocities results in very long times during which the coupled systems exhibit alternatively irrational phase synchronization and random phase diffusion. The range of parameters for which the phenomenon occurs contains an open set, and is thus physically observable.

Dynamics of multiple myeloma tumor therapy with a recombinant measles virus.

Replication-competent viruses are being tested as tumor therapy agents. The fundamental premise of this therapy is the selective infection of the tumor cell population with the amplification of the virus. Spread of the virus in the tumor ultimately should lead to eradication of the cancer. Tumor virotherapy is unlike any other form of cancer therapy as the outcome depends on the dynamics that emerge from the interaction between the virus and tumor cell populations both of which change in time. We explore these interactions using a model that captures the salient biological features of this system in combination with in vivo data. Our results show that various therapeutic outcomes are possible ranging from tumor eradication to oscillatory behavior. Data from in vivo studies support these conclusions and validate our modeling approach. Such realistic models can be used to understand experimental observations, explore alternative therapeutic scenarios and develop techniques to optimize therapy.

Branching dendrites with resonant membrane: a "sum-over-trips" approach.

Dendrites form the major components of neurons. They are complex branching structures that receive and process thousands of synaptic inputs from other neurons. It is well known that dendritic morphology plays an important role in the function of dendrites. Another important contribution to the response characteristics of a single neuron comes from the intrinsic resonant properties of dendritic membrane. In this paper we combine the effects of dendritic branching and resonant membrane dynamics by generalising the "sum-over-trips" approach (Abbott et al. in Biol Cybernetics 66, 49-60 1991). To illustrate how this formalism can shed light on the role of architecture and resonances in determining neuronal output we consider dual recording and reconstruction data from a rat CA1 hippocampal pyramidal cell. Specifically we explore the way in which an Ih current contributes to a voltage overshoot at the soma.

Local bifurcations in the symmetric model of selection with fertility differences.

The analysis of difference equations in population dynamics is frequently confined to the calculation of fixed points and the determination of their stability. In general it is not possible to deduce much about the global behavior of the system from such information. This paper presents a study of the bifurcations of fixed points in the symmetric model of selection with fertility differences. Equivalent methods can be applied to other continuous and discrete models to study the dynamics in the vicinity of fixed points in a systemic way and as a first step in an analysis of the global dynamics.