Scale-Free Navigational Planning by Neuronal Traveling Waves

Abstract

Spatial navigation and planning is assumed to involve a cognitive map for evaluating trajectories towards a goal. How such a map is realized in neuronal terms, however, remains elusive. Here we describe a simple and noise-robust neuronal implementation of a path finding algorithm in complex environments. We consider a neuronal map of the environment that supports a traveling wave spreading out from the goal location opposite to direction of the physical movement. At each position of the map, the smallest firing phase between adjacent neurons indicate the shortest direction towards the goal. In contrast to diffusion or single-wave-fronts, local phase differences build up in time at arbitrary distances from the goal, providing a minimal and robust directional information throughout the map. The time needed to reach the steady state represents an estimate of an agent's waiting time before it heads off to the goal. Given typical waiting times we estimate the minimal number of neurons involved in the cognitive map. In the context of the planning model, forward and backward spread of neuronal activity, oscillatory waves, and phase precession get a functional interpretation, allowing for speculations about the biological counterpart.

Fig 1. A comparison of the classical diffusion model and our scale-free traveling wave model in 1 dimension.

For illustration, a linear environment with 20 discrete neurons and goal at position 1 is considered. In the diffusion model (a), activity (membrane voltage) is spread from the goal across the environment (top panel) with exponentially decaying gradient, and hence quickly fading directional information (bottom). In the traveling wave model (b), activation of the goal synaptically spreads through the environment such that the firing phase of adjacent neurons increases linearly with distance from the goal (top), resulting in a fixed and positive local phase difference and hence in directional information that does not decay in space and is not restricted to a specific scale (‘scale-free’), bottom.

Fig 2. Network architecture, traveling waves and planning time.

(a) Planning and readout network. For each neuron in the planning layer, 4 actions can be assigned. Actions neurons associated to planning neuron (i, j) are W _ij, E _ij, N _ij and S _ij which receive synaptic input respectively from the left, right, north, and south neighbor of the neuron (i, j) and evoke a motion in the same directions (just one synaptic input is shown). Action neurons corresponding to the current place of the agent (here again (i, j)) are driven by an additional input ($I_{ext}^{a_{ij}}$). The first of the 4 action neurons that is fired by the passing traveling wave inhibits the other 3 action neurons. (b) Synaptically propagating waves of activity from the goal neuron at (1,1) across a planning layer of 20 × 20 neurons, for four different obstacle configurations. Colors code for firing phases at steady state relative to the goal neuron. (c) Time courses of the local phase difference for two sample neurons at positions (10,10) and (20,20), indicated by □ and ○ in the top left panel of b, with their local west-positioned neighbors. The time to reach the maximal local phase difference represents the planning time for these two start positions towards the goal (here 600 and 1050 ms, vertical lines), and subsequently the full path towards the goal can be read out.

Fig 3. Results in a network of 10 × 10 planning neurons with different noise scenarios.

Start position at (7,9), goal position at (7,3) and obstacle indicated by the black bar. Top row: voltage of the four nearest neighbor neurons at the start position (identity color coded). Middle row: external input current these neurons receive. Bottom row: 2D environment with the chosen path (blue line) from the start to the goal. Standard deviations σ _ext of the input currents indicated below. a, b: The mean of the input current generated by the background Poisson spiking neuron was constant and identical for all neurons (μ _ext = 12 mV/ms), except for the goal neuron (marked with g, ${\mu }_{ext}^{goal}=12.5$). c: A common sinusoidal fluctuation in the Poisson firing rate of the background neurons does not disturb the relative timing among neighboring neurons. d: A randomly chosen bias in the mean ${\mu }_{ext}^{ij}$ of the individual input currents with standard deviation 3 does not prevent the agent from finding a short path to the goal. In all simulations, the planning times were 600 ms, the readout times 250 ms, and the coupling strength was ϵ = 0.15.

Fig 4. Effect of noise on planning performance and readout time in the network used in Fig 3.

(a) Planning performance, shown for 3 different readout times of 60, 180 and 240 ms (corresponding to 1, 3 and 4 readout cycles, bottom to top), declines with increasing noise (average across 10 chosen paths, error bars represent standard deviations of mean). (b) Readout time used at each position such that a shortest 10-step path is found, evaluated for the different noise levels. Parameters, network- and task configuration as used in Fig 3.

Fig 5. Planning performance decreases with decreasing oscillation frequency and is worst for a solitary wave.

(a) Planning performance as a function of the noise level for a 18 Hz (blue) and a 11 Hz (green) intrinsic oscillation frequency with 4 readout cycles to select a single action. For the solitary wave, planning performance was measured after averaging the path lengths across 4 sweeps. Error bars from 10 realizations. (b, c) Color coded spike times relative to the spike time of the goal position (1,1) for (b) the 18 Hz periodic traveling wave after reaching steady state and (c) the solitary wave, both with noise level σ _ext = 0.2. For the solitary wave, the spike times do not faithfully represent distance from goal, and hence the action selection mechanism may yield a path to a non-goal position (the blue island in (c)).

Fig 6. Determinants of the planning time.

(a) Snapshots of the local phase differences along diagonal positions from the goal. The intrinsic oscillation frequency was 17 Hz for the non-goal positions and 18 Hz for the goal position. Noise level was σ _ext = 0. (b) Planning time (i.e. the time to reach roughly 90% of the final local phase difference) increases linearly with the distance from the goal. (c) Planning time also increases with the frequency of the intrinsic oscillation.

Fig 7. Finding shortcut and adaptive planning.

(a, b) Turning up the noise level in the external input (indicated below each column) may be a way to prevent the detection of a shortest path. Other parameters as indicated in the caption of Fig 3. (c) Adaptive planning for a moving goal. A goal at initial goal position (17,7) is moved along the red line after planning time is over and the agent heads from the start towards this goal position. The goal moves one step at each readout cycle. The moving target can reliably be traced (blue line). The planning time is 1 s and the readout time is 250 ms. ${\mu }_{ext}^{goal}=12.5$, μ _ext = 12.

Fig 8. Planning in complex environments.

(a) Propagated wave of activity from the goal at (1,1) through a 20 × 20 network with obstacles (black bars) is demonstrated in space-space color coded plot of firing times relative to the goal neuron. The intrinsic frequency of the goal neuron is 18 Hz. (b, c) Two examples of navigation path (blue lines) in the network from different start positions (•) to the same goal (◂) such that a slight shift of the starting position implies an entirely different shortest path. The input to the planning layer is ${\mu }_{ext}^{goal}$ = 12.5 and μ _ext = 12 and the noise level is σ _ext = 0.7. The planning time takes 1 s and readout time is 150 ms.