Mark M. Millonas

How Swarms Build Cognitive Maps

Swarms of social insects construct trails and networks of regular traffic via a process of pheromone laying and following. These patters constitute what is known in brain science as a cognitive map. The main difference lies in the fact that the insects write their spatial memories in the environment, while the mammalian cognitive map lies inside the brain. This analogy can be more than a poetic image, and can be further justified by a direct comparison with the neural processes associated with the construction of cognitive maps in the hippocampus. We investigate via analysis and numerical simulation the formation of trails and networks in a collection of insect-like agents. The agents interact in simple ways which are determined by experiments with real ants.

Transport and current reversal in stochastically driven ratchets

Abstract We present analytic results for the current in a system moving in an arbitrary periodic potential and driven by weak Gaussian noise with an arbitrary power spectrum which are valid to order ( t c t r ) 2 , where t c is the largest characteristic time of the noise, and t r is the characteristic intrawell relaxation time. The dependence of the current on the shape of the potential, and on the shape of the power spectrum of the noise is illustrated. It is demonstrated that the direction of the current is opposite when the power spectrum of the noise has a minimum or maximum at zero frequency. A simple physical mechanism for this behavior is suggested. The behavior of the system in the limit of slow noise ( t c ⪢ t r ) is also discussed.

Pattern Formation and Functionality in Swarm Models

We explore a simplified class of models we call swarms, which are inspired by the collective behavior of social insects. We perform a mean-field stability analysis and numerical simulations of the model. Several interesting types of behavior emerge in the vicinity of a second-order phase transition in the model, including the formation of stable lines of traffic flow, and memory reconstruction and bootstrapping. In addition to providing an understanding of certain classes of biological behavior, these models bear a generic resemblance to a number of pattern formation processes in the physical sciences.

A connectionist type model of self-organized foraging and emergent behavior in ant swarms*

A model of how the local scent laying and scent following of individual ants leads to collective decision making in groups of ants is studied. We confine ourselves here to ants whose motion is restricted to a set of n paths, or one-dimensional segments, connected in an arbitrary way to other segments forming a network. These networks are analogous to some recent laboratory experiments with ants. A microscopic approach is used, where the behavior of the individual ants is completely described by a local choice function based on their observed local behavior. This function gives the allowed transition probabilities to go from a given segment to a final segment in terms of the scents on these final segments. The ants are shown to be analogous to particles with n energy states in equilibrium with a heat bath of a given temperature. The energies of these states evolve according to a set of non-linear differential equations. The various self-organizing and emergent behaviors of groups of ants can then be viewed as instabilities in the dynamics of these equations. The simplest cases where the ants collectively choose between routes of equal and unequal length, and between food sources of varying quality, are examined in detail, and comparisons made with the experimental results ( Pasteels et al. , 1987 a ; Deneubourg et al. , 1990 ; Goss et al. , 1990 ). Such ant networks are connectionist type models whose architecture and dynamics parallel those of many interesting systems such as neural networks, autocatalytic chemical reactions ( Farmer et al. , 1986 b ), classifier systems, and immune networks ( Farmer, 1990) .

Observable and hidden singular features of large fluctuations in nonequilibrium systems

We study local features, and provide a topological insight into the global structure of the probability density distribution and of the pattern of the optimal paths for large rare fluctuations away from a stable state. In contrast to extremal paths in quantum mechanics, the optimal paths do not encounter caustics. We show how this occurs, and what, instead of caustics, are the experimentally observable singularities of the pattern. We reveal the possibility for a caustic and a switching line to start at a saddle point, and discuss the consequences.

Nonlinear statistical modeling and model discovery for cardiorespiratory data.

We present a Bayesian dynamical inference method for characterizing cardiorespiratory (CR) dynamics in humans by inverse modeling from blood pressure time-series data. The technique is applicable to a broad range of stochastic dynamical models and can be implemented without severe computational demands. A simple nonlinear dynamical model is found that describes a measured blood pressure time series in the primary frequency band of the CR dynamics. The accuracy of the method is investigated using model-generated data with parameters close to the parameters inferred in the experiment. The connection of the inferred model to a well-known beat-to-beat model of the baroreflex is discussed.

Recovering ‘lost’ information in the presence of noise: application to rodent–predator dynamics

A Hamiltonian approach is introduced for the reconstruction of trajectories and models of complex stochastic dynamics from noisy measurements. The method converges even when entire trajectory components are unobservable and the parameters are unknown. It is applied to reconstruct nonlinear models of rodent–predator oscillations in Finnish Lapland and high-Arctic tundra. The projected character of noisy incomplete measurements is revealed and shown to result in a degeneracy of the likelihood function within certain null-spaces. The performance of the method is compared with that of the conventional Markov chain Monte Carlo (MCMC) technique.

Cooperative Phenomena in Swarms

A model of the cooperative behavior of a large number of locally acting organisms is proposed. The space in which the organisms move is discretized, and is modeled by a lattice of nodes, or cells. Each cell has a specified volume, and is connected to other cells in the space in a definite way. Organisms move probabilistically between local cells in this space, but with weights dependent on local morphogenic substances, or morphogens. The morphogens are in turn are effected by the passage of an organism. The evolution of the morphogens, and the corresponding flow of the organisms constitutes the collective behavior of the group. The generic properties of such systems are analyzed, and a number of results are obtained. The model has various types of phase transitions and self-organizing properties controlled both by the level of the noise, and other parameters.

Applications of dynamical inference to the analysis of noisy biological time series with hidden dynamical variables

We present a Bayesian framework for parameter inference in noisy, non‐stationary, nonlinear, dynamical systems. The technique is implemented in two distinct ways: (i) Lightweight implementation to be used for on‐line analysis, allowing multiple parameter estimation, optimal compensation for dynamical noise, and reconstruction by integration of the hidden dynamical variables, but with some limitations on how the noise appears in the dynamics; (ii) Full scale implementation of the technique with extensive numerical simulations (MCMC), allowing for more sophisticated reconstruction of hidden dynamical trajectories and dealing better with sources of noise external to the dynamics (measurements noise).

Parameter and Structure Inference for Nonlinear Dynamical Systems

A great many systems can be modeled in the nonlinear dynamical systems framework, as ẋ = f(x) + ξ(t), where f() is the potential function for the system, and ξ is the excitation noise. Modeling the potential using a set of basis functions, we derive the posterior for the basis coefficients. A more challenging problem is to determine the set of basis functions that are required to model a particular system. We use the Bayesian Information Criteria (BIC) to rank models, together with the beam search to search the space of models. We show that we can accurately determine the structure of simple nonlinear dynamical system models, and the structure of the coupling between nonlinear dynamical systems where the individual systems are known. This last case has important ecological applications.