Modelling of excitation propagation on computer models of insoles colonised by fungal mycelium. Videos and potential difference recordings.
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We used an artistic image of the mycelium network projected onto a $364 \times 985$ nodes grid. <br> The original image $M=(m_{ij})_{1 \leq j \leq n_i, 1 \leq j \leq n_j}$, $m_{ij} \in \{ r_{ij}, g_{ij}, b_{ij} \}$, where $n_i=364$ and $n_j=985$, and $1 \leq r, g, b \leq 255$, was converted to a conductive matrix $C=(m_{ij})_{1 \leq i,j \leq n}$ derived from the image as follows: $m_{ij}=1$ if $r_{ij}>170$, $g_{ij}>170$ and $b_{ij}<200$; a dilution operation was applied to $C$. FitzHugh-Nagumo (FHN) equations is a qualitative approximation of the Hodgkin-Huxley model of electrical activity of living cells:<br> \begin{eqnarray}<br> \frac{\partial v}{\partial t} & = & c_1 u (u-a) (1-u) - c_2 u v + I + D_u \nabla^2 \\<br> \frac{\partial v}{\partial t} & = & b (u - v),<br> \end{eqnarray}<br> where $u$ is a value of a trans-membrane potential, $v$ a variable accountable for a total slow ionic current, or a recovery variable responsible for a slow negative feedback, $I$ {is} a value of an external stimulation current. The current through intra-cellular spaces is approximated by<br> $D_u \nabla^2$, where $D_u$ is a conductance. The term $D_u \nabla^2 u$ governs a passive spread of the current. The terms $c_2 u (u-a) (1-u)$ and $b (u - v)$ describe the ionic currents. The term $u (u-a) (1-u)$ has two stable fixed points $u=0$ and $u=1$ and one unstable point $u=a$, where $a$ is a threshold of an excitation. We integrated the system using the Euler method with the five-node Laplace operator, a time step $\Delta t=0.015$ and a grid point spacing $\Delta x = 2$, while other parameters were $D_u=1$, $a=0.13$, $b=0.013$, $c_1=0.26$. We controlled excitability of the medium by varying $c_2$ from 0.05 (fully excitable) to 0.015 (non excitable). Boundaries are considered to be impermeable: $\partial u/\partial \mathbf{n}=0$, where $\mathbf{n}$ is a vector normal to the boundary. To record dynamics of excitation in the network, as if in laboratory experiments, we simulated electrodes by calculating a potential $p^t_x$ at an electrode location $x$ as $p_x = \sum_{y: |x-y|<2} (u_x - v_x)$. Configuration of electrodes $1, \cdots, 16$ is shown in Fig.~\ref{fig:mycelium}c. Time-lapse snapshots provided in the paper were recorded at every 100\textsuperscript{th} time step, and we display sites with $u >0.04$; videos and figures were produced by saving a frame of the simulation every 100\textsuperscript{th} step of the numerical integration and assembling the saved frames into the video with a play rate of 30 fps. Insole_01: Excitation started at electrode E2 Insole_10: Excitation started at electrode E1 Insole_11: Excitation started at electrodes E1 and E2
