A model of a topologically organized neural network of a Hop.field type with nonlinear analog neurons is shown to be very effective for path planning and obstacle avoidance. This deterministic system can rapidly provide a proper path, from any arbitrary start position to any target position, avoiding both static and moving obstacles of arbitrary shape. The model assumes that an ( external) input activates a target neuron, corresponding to the target position, and specifies obstacles in the topologically ordered neural map. The path .follows from the neural network dynamics and the neural activity gradient in the topologically ordered map. The analytical results are supported by computer simulations to illustrate the performance of the network.

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We study the critical properties of a self-avoiding walk on percolation clusters using exact enumerations, in both d=2 and d=3. Calculations of the exponent g, which measures free-energy fluctuations, clearly show the existence of two distinct phases, one for strong disorder (at the percolation threshold), the other for weak disorder (above the percolation threshold). The v exponent is, within numerical accuracy, the same in the two phases. However in d=3, its value, v=0.64~0.015 is significantly larger than on a pure lattice.PACS number(s): 64.60.Ak, 36.20.Ey

\ye study on-line learning with a momentum term for nonlinear leaming rules. Through introduction of auxiliary variables, we show that the leaming process can be desc"bed by a Markov process. For small leming pxameters q and momentum parmeten (I close to I, such that y = ,,/(I-(I)? is finite. the timescales for the evolution of the weights and the auxiliary variables we the same. In this m e Van Kampen's expansion can be applied in a straightfoward m n e r. We obtain evolutian equations for the average network state and the Eucmstions around this average. These evolution equations depend (after reSWling a of the time and fluctuations) only on y : all combinakions (q , (I) with the same value of y give rise to similar behaviour. The wse with (I constant and q small requires a completely different analysis There are two different timescales: a fast timescale on which the auxiliary variables equilibrate and a slow timescale for the change of the weighs. By projection on the space of slow variables the fast variables can be eliminated. We hnd that, for small leaming parameters q and finite momen" parameters (I, lewning with momentum is equivalent to leaming without a momentum term with a rescaled learning parameter 6 = q / (l-(I). Simulations with the nonlinear Oja learning rule confirm the theoretical resulll.

The basins of attraction of extremely diluted neural-network models in the presence of external neural stimuli parallel to the starting configuration are calculated analytically. For moderate values of the storage capacity a, the basins of attraction can be enlarged significantly. For larger values of a, the patterns are still locally stable but become dynamically blocked by the external stimuli so that the effective storage capacity decreases. The performance can be improved further by allowing for time-dependent stimuli.

In this paper we present a biologically inspired two-layered neural network for trajectory formation and obstacle avoidance. The two topographically ordered neural maps consist of analog neurons having continuous dynamics. The first layer, the sensory map, receives sensory information and builds up an activity pattern which contains the optimal solution (i.e. shortest path without collisions) for any given set of current position, target positions and obstacle positions. Targets and obstacles are allowed to move, in which case the activity pattern in the sensory map will change accordingly. The time evolution of the neural activity in the second layer, the motor map, results in a moving cluster of activity, which can be interpreted as a population vector. Through the feedforward connections between the two layers, input of the sensory map directs the movement of the cluster along the optimal path from the current position of the cluster to the target position. The smooth trajectory is the result of the intrinsic dynamics of the network only. No supervisor is required. The output of the motor map can be used for direct control of an autonomous system in a cluttered environment or for control of the actuators of a biological limb or robot manipulator. The system is able to reach a target even in the presence of an external perturbation. Computer simulations of a point robot and a multi-joint manipulator illustrate the theory.

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In this study we investigate the time evolution of the activity in a topographically ordered neural network with external input for two types of neurons: one network with binaryvalued neurons with a stochastic behaviour and one with deterministic neurons with a continuous output. We demonstrate that for a pariicular range of lateral intemtion strengths, changes in external input give rise to gradual changes in the position of clustered neural activity.The theoretical results are illusuared by computer simulations in which we have simulared a neural network model for trajectory planning for a multi-joint maniplotor. The model gives a collision-free trajectory by combining the sensory information about the position of target and obstacles. The position of the manipulator is uniquely related to the clustered activity of the population of neurons, the population vector. The movement of the manipulator from any initial position to the target position is the result of the intrinsic dynamics of the network.