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On diversity of configurations generated by excitable cellular automata with dynamical excitation intervals

Excitable cellular automata with dynamical excitation interval exhibit a wide range of space-time dynamics based on an interplay between propagating excitation patterns which modify excitability of the automaton cells. Such interactions leads to formation of standing domains of excitation, stationary waves and localised excitations. We analysed morphological and generative diversities of the functions studied and characterised the functions with highest values of the diversities. Amongst other intriguing discoveries we found that upper boundary of excitation interval more significantly affects morphological diversity of configurations generated than lower boundary of the interval does and there is no match between functions which produce configurations of excitation with highest morphological diversity and configurations of interval boundaries with highest morphological diversity. Potential directions of future studies of excitable media with dynamically changing excitability may focus on relations of the automaton model with living excitable media, e.g. neural tissue and muscles, novel materials with memristive properties, and networks of conductive polymers.


Introduction
Since their popularisation in [5], excitable cellular automata became a convenient tool for studying complex phenomena of excitation dynamics and chemical reaction-diffusion activities in physical, chemical and biological systems [7,3]. The cellular automata offers quick 'prototyping' of complex spatially extended non-linear media. The examples of 'best practice' include models of Belousov-Zhabotinsky reactions and other excitable systems [4,8], chemical systems exhibiting Turing patterns [12,9,10], precipitating systems [2], calcium wave dynamics [11], and chemical turbulence [6].
In a classical Greenberg-Hasting [5] automaton model of excitation a cell takes three states -reseting, excited and refractory. A resting cell becomes excited if number of excited neighbours exceeds a certain threshold, an excited cell becomes refractory, and a refractory cell returns to its original resting state. In 1998 [1], we introduced an excitable cellular automaton, where a resting cell is excited if a number of its excited neighbours belongs to some fixed interval [θ 1 , θ 2 ]. The interval [θ 1 , θ 2 ] was called an excitation interval. For a two-dimensional cellular automaton with eight-cell neighbourhood boundaries of the excitation interval satisfy the condition: 1 ≤ θ 1 ≤ θ 2 ≤ 8. We found that by tuning θ 1 and θ 2 we can force the automaton to imitate almost all kinds of excitation dynamics, from classical target and spiral waves observed in physical and chemical excitable media to wave-fragments inhabiting sub-excitable media.
How does excitation dynamics change if we allow boundaries of the excitation interval to change during the automaton development? We partially answer the question in present paper by making the interval [θ t 1 (x), θ t 2 (x)] of every cell x to be dynamically updatable at every step t depending on state of the cell x and numbers of excited and refractory neighbours in the cell x's neighbourhood.
The excitable automata with dynamical excitation intervals are defined in Sect. 2. Morphological diversity of cellular automata (measured using Shannon entropy and Simpson index) with different functions of interval updates is characterised in Sect. 3. Section 4 characterises generative diversity (measured in terms of different configurations generated during space-time development of automaton starting with a single non-resting cell) of the local transitions. Some afterthoughts are offered in Sect. 5.

Dynamical excitation intervals
Let x t and x t+1 be states of a cell x at time steps t and t + 1, and σ t + (x) be a sum of excited neighbours in cell x's neighbourhood u(x) = {y : |x − y| L∞ = 1}. Cell x updates its state by the following rule: The boundaries θ t 1 (x) and θ t 2 (x) are dynamically updated depending on cell x's state and numbers of x's excited σ t + (x) and refractory σ t − (x) neighbours. A natural way to update boundaries is by increasing or decreasing their values as follows: and -1 if a < b, and ξ(a) = 1 if a < 1 and 8 if a > 8. Boundaries of excitation interval [θ t 1 (x), θ t 2 (x)] are updated independently of each other. Local excitation rules are determined by values of T 1 , · · · T 4 . We therefore address the functions as tuples E( Functions E(a0b0), a ∈ {−1, 0, 1} represent rules with fixed upper boundary θ 2 of excitation and dynamically updated lower boundary θ 1 . These are equivalent to dynamically updated thresholds of excitation. Functions E(0a0a), a ∈ {−1, 0, 1} represent rules with fixed lower boundary and dynamical upper boundary of excitation interval.
The experiments are conducted on a cellular array of n×n cells with periodic boundary conditions. In a typical experiment we perturb resting cellular array with a localised domain of excitation, wait till transient period is over (1000 iterations is enough) and most excitation patterns collide and disappear and persist indefinitely, and then analyse three configurations: configuration of excitation represented by an array of cells states x t , and configurations of lower θ t 1 (x) and upper θ t 2 (x) boundaries of excitation intervals. Initially θ 0 2 (x) = 8 for any x. In experiments we considered initial conditions θ 0 1 (x) = 1 and θ 0 1 (x) = 2. The following scenaria of initial excitation were played: • (++)-start, θ 1 = 2: all cells are resting but two neighbouring cells are assigned excited state, • R1-start: let D be a disc radius n/4 centred in the array L of n × n cells, all cells are resting but cells lying in D are assigned excited states with probability 0.2 and θ 0 1 (x) = 1 for any x, • R2-start: all cells are resting but cells lying in D get excited states with probability 0.2 and θ 0 1 (x) = 2 for any x, • D1-start: all cells are resting but cells lying in D get excited states with probability 0.1 or refractory states with probability 0.1 and θ 0 1 (x) = 1 for any x, • D2-start: all cells are resting but cells lying in D get excited states with probability 0.1 or refractory states with probability 0.1 and θ 0 1 (x) = 2 for any x, • (−+)-start: all cells are resting but one cell is excited and its western neighbour is refractory, • (− + +)-start all cells are resting but one cell is excited, its first order western neighbour is excited and its second order western neighbour is refractory. Cell states were represented by colours and grey levels as follows: excited state + is red (c. 76 grey), resting state is white and refractory state − is blue (c. 28 grey). Colour values of excitation interval boundaries θ 1 and θ 2 are following: 1 is white, 2 is green or 150 grey, 3 is yellow or 226 grey, 4 is blue or 28 grey, 5 is magenta or 104 grey, 6 is cyan or 178 grey, 7 is red or 76 grey, and 8 is black.

Morphological diversity
We evaluated morphological diversity of configurations of excitation and using Shannon entropy and Simpson's index. Let W = {•, +, −} be a set of all possible configurations of a 9-cell neighbourhood w(x) = u(x) ∪ x, x ∈ L. Let c be a configuration of automaton, we calculate number of non-resting neighbourhood configurations as η = x∈L (x), where (x) = 0 if for every resting x all its neighbours are resting, and (x) = 1 otherwise. The Shannon entropy is calculated as E = − w∈W (ν(w)/η · ln(ν(w)/η)), where ν(w) is a number of times the neighbourhood configuration w is found in automaton configuration c. Simpson's index is calculated as  . Morphological diversity of functions for D1-start, θ 0 1 (x) = 1 for all x: Shannon entropy E 1 for configuration of θ 1 (horizontal axis) vs Shannon entropy E 2 for configuration of θ 2 (vertical axis), recorded at t = 1000. Encircled data points are seven functions with highest morphological diversity specified in column D1-start, E 1 -E 2 in Tab. 3.    Fig. 1-4, are grouped in Tab. 3 for various scenarios of initial start.
. The boundary θ 2 (x) decreases if cell x conforms to excitation-refractoriness ratio in its neighbourhood: x t = + and σ t . Increase of θ 1 and decrease of θ 2 lead to decrease cell's excitability. Thus we can characterise function E(1 − 101) as follows: excitability of a cell decreases if the cell dissents with its neighbourhood and increases otherwise.     decreases if cell x is excited and has more refractory than excited neighbours. Excitability of a cell decreases if the cell dissents with excitation ratio in its neighbourhood, and increases otherwise.      There are only two functions, E(1010) and E(1000), where only lower boundary θ 1 is updated in Tab. 3. Function E(1010) represents a situation when θ 1 (x) is independently of a state of cell x: θ 1 (x) increases if number of excited neighbours exceeds number of refractory neighbours, and θ 1 (x) decreases if refractory neighbours outnumber excited neighbours. In automata governed by function E(1000) value of θ 1 (x) is updated as above but only if cell x is excited. There are several functions with highest morphological diversity which represent fixed lower boundary and dynamical upper boundary, e.g. E(000 − 1) (Figs. 5a and 10a), E(0100) (Figs. 6a and 8a), E(0101) (Fig. 7a), E(0 − 100), E(0 − 101), and E(0001) (Tab. 3).

Generative diversity and localisations
Generative diversity characterises how many different configurations are generated during space-time development of automaton starting with entirely resting configuration but single cell in a non-resting state. We consider two starting conditions: (++)-start      See Tabs. 2 and 3. Configurations generated by function E(11 − 10) are exemplified in Fig. 12b and function E(00 − 1 − 1) in Fig. 13a. Functions E(100 − 1) and E(00 − 1 − 1) are the functions with minimal updates of excitation interval boundaries. In automata governed by function E(00 − 1 − 1) θ 1 (x) and θ 2 (x) are updated only if the cell x is in refractory state: both boundaries decrease if excited neighbours of x outnumber refractory neighbours, and they increase refractory neighbours dominate. In automata governed by function E(100 − 1) the boundary θ 1 (x) increases if the excited cell x has more excited neighbours than refractory ones, and boundary and decreases if the cell has more refractory neighbours. The boundary θ 2 (x) is updated only if cell x is refractory: θ 2 (x) decreases if excited neighbours outnumber refractory neighbours, and it increases otherwise.

Summary
Excitable cellular automata with dynamical excitation interval exhibit a wide range of space-time dynamics based on an interplay between propagating excitation patterns which modify excitability of the automaton cells. Such interactions leads to formation of standing domains of excitation, stationary waves and localised excitations. We analysed morphological and generative diversities of the functions studied and characterised the functions with highest values of the diversities. Amongst other intriguing discoveries we found that upper boundary of excitation interval more significantly affects morphological diversity of configurations generated than lower boundary of the interval does and there is no match between functions which produce configurations of excitation with highest morphological diversity and configurations of interval boundaries with highest morphological diversity. Potential directions of futures studies of excitable media with dynamically changing excitability may focus on relations of the automaton model with living excitable media, e.g. neural tissue and muscles, and novel materials with memristive properties, and networks of conductive polymers.