A minimal model of cognition based on oscillatory and current-based reinforcement processes

TL;DR

This model demonstrates how a combination of oscillatory and current-based reinforcement processes can be used to couple resources in an efficient manner, mimicking the way these organisms function, and discusses connections between this model and basal cognition in biological systems and slime moulds, in particular, how oscillations might contribute to self-organized problem-solving by these organisms.

Abstract

Building mathematical models of brains is difficult because of the sheer complexity of the problem. One potential starting point is through basal cognition, which gives abstract representation of a range of organisms without central nervous systems, including fungi, slime moulds and bacteria. We propose one such model, demonstrating how a combination of oscillatory and current-based reinforcement processes can be used to couple resources in an efficient manner, mimicking the way these organisms function. A key ingredient in our model, not found in previous basal cognition models, is that we explicitly model oscillations in the number of particles (i.e. the nutrients, chemical signals or similar, which make up the biological system) and the flow of these particles within the modelled organisms. Using this approach, we find that our model builds efficient solutions, provided the environmental oscillations are sufficiently out of phase. We further demonstrate that amplitude differences can promote efficient solutions and that the system is robust to frequency differences. In the context of these findings, we discuss connections between our model and basal cognition in biological systems and slime moulds, in particular, how oscillations might contribute to self-organised problem-solving by these organisms.

Figures (15)

• Figure 1: The model on a $C_5$ graph, where $l_{ij}$ is the length of the edge between node $i$ and $j$, $D_{ij}$ is the conductivity (thickness) of the edge between $i$ and $j$. In (a) the nodes are non-oscillating, and node 1 is a source, with an inflow at rate $a$ and node 3 is a sink with an outflow at rate $bN_3$. In (b) the nodes are oscillating as $A_1(1 + \sin(2\pi\theta_1t + \phi_1))- bN_1(t)$ and $A_3(1 + \sin(2\pi\theta_3t + \phi_3))$$- bN_3(t)$ at node 1 and 3 respectively, meaning that the different colours of the oscillating nodes indicates different amplitude, frequencies and phases in the oscillators.
• Figure 2: Example of a large regular graph of degree 3, where the node in blue corresponds to an oscillatory node with phase $0$ and the nodes in red correspond to oscillatory nodes with phase $\pi$ in later simulations.
• Figure 3: Results from the model on a $C_5$ graph, with node $1$ as a source, node $3$ as a sink, an inflow at rate $a=1$, an outflow at rate $b=0.5$, and the lengths of all edges are $l_{ij} = 10$ apart from $l_{12}$. On the left, we set $l_{12} = 10$, thus the path going through 1-2-3, is the shortest one. On the right, we set $l_{12} = 30$, thus the path going through 1-5-4-3 is the shortest one. The first row shows the input at node $1$ and output at node $3$, the second row plots the number of particles on each node, $N_i(t)$, the third row indicates the conductivity of the edges in the shortest path, and the last row indicates the conductivity of the remaining edges in the graph, where the dashed lines in the corresponding colour show the analytical results of the steady state.
• Figure 4: Bifurcation diagrams with $l_{12}$ as bifurcation parameter. Here $l_{23}=l_{34}=l_{45}=l_{51}=10$, $a=1.0$, $b=0.5$, $q=0.1$ and $\lambda=0.01$. The left plot shows the number of particles at each node, while the right plot displays the conductivity $D_{(i,j)}$ of the edges, both as a function of $l_{12}$. The diagrams are generated by simulating the equation system \ref{['eq:C_5_eq_steady']} through 100,000 time steps, each time initiated with random initial conditions. This process is repeated 10 times for each value of $l_{12}$, with the bifurcation diagram displaying the results from the last 100 time steps of each simulation
• Figure 5: Different equilibrium points for the model on the $C_5$ graph with non-oscillatory sources and sinks, where node 1 (blue) is a source, and node 3 (green) is a sink. In (a) we see the stable equilibrium when 1-2-3 is the shortest path, in (b) we see the stable equilibrium when both paths have equal length, and in (c) we see the stable equilibrium when 1-5-4-3 is the shortest path.