Pulsing dynamics in randomly wired glider cellular automata
PPulsing dynamics in randomly wiredglider cellular automata
Andrew Wuensche ∗ Discrete Dynamics Lab.
Edward Coxon † Dept. of Anaethesia and Pain Medicine,The Canberra Hospital, ACT, Australia.
Abstract
Sustained rhythmic oscillations, pulsing dynamics, emerge spontaneouslywhen the local connection scheme is randomised in 3-value cellular au-tomata that feature“glider” dynamics. Time-plots of pulsing measuresmaintain a distinct waveform for each glider rule, and scatter plots of en-tropy/density and the density return-map show unique signatures, whichhave the characteristics of chaotic strange attractors. We present casestudies, possible mechanisms, and implications for oscillatory networks inbiology. keywords: cellular automata, glider dynamics, random wiring, pulsing,bio-oscillations, emergence, chaos, complexity, strange attractor, heartbeat,sympathetic centre, central pattern generator
Arguably the most interesting manifestation of cellular automata dynamics isthe emergence of mobile (and stable) configurations known as particles or glid-ers which interact by collisions, possibly making compound emergent structuressuch as glider-guns in an open ended hierarchy — components which can some-times be rearranged to achieve universal computation[1, 12]. Glider dynamicsarise within rare “complex” rules, which also include dynamics apart from glid-ers, for example, dynamic patches, blinkers, or mobile boundaries between do-mains. Otherwise the dynamics and rule types, broadly speaking, are either “or-dered” or “disordered” judged by subjective impressions of space-time patterns,but also by objective measures such as input-entropy and its variability[31],basin of attraction topology[30], and Derrida plots[9, 37]. By any evaluation,disorder comprises the vast majority of large rule-spaces.We pose the question: while preserving a homogeneous rule, what kindof dynamics would result if the regular local neighborhood connections (thewiring) of classical CA are randomised? — an experiment readily implementedin DDLab[37, 38], with its functionality for toggling between regular and random ∗ [email protected], † [email protected] a r X i v : . [ n li n . C G ] A ug iring on-the-fly, and where random wiring can be fully random or confined ina local zone, or even re-randomised at each time-step.The results of these “random wiring” experiments reveal a novel and remark-able phenomenon — for 3-value, k-totalistic, “glider” CA rules, sustained rhyth-mic oscillations, pulsing behaviour, is the inevitable outcome. The pulsing is ob-vious to the subjective eye when observing space-time patterns, but is also char-acterised by objective measures: the density of each value across the network,and the collective input-entropy. Time-plots of pulsing measures for each gliderrule maintain a particular wavelength ( wl ), wave-height ( wh , twice amplitude),and waveform (its shape or phase), and scatter plots of entropy/density[31] andthe density return-map[37] show distinct signatures, which have the character-istics of chaotic strange attractors. We will use the term “waveform” to sumup these pulsing measures, and the “CA pulsing model” for the system itself.We demonstrate pulsing when the wiring is fully (and sometimes partly) ran-domised. Pulsing is robust to re-randomised wiring at each time-step, to noise,to boundary conditions, and to asynchronous or sequential updating. Whenrandom wiring is confined in a relatively small local zone, spiral density waves,reminiscent of reaction-diffusion, can emerge in a large enough system, so localpulsing is still present (figure 14) as waves sweep over local areas of the lattice.Experiment shows that pulsing does not occur for ordered or disorderedrules, or for complex rules that do not feature well defined gliders. Pulsing isnot discernible for glider rules in binary CA, such as the 1D rule 110, the 2DGame-of-Life[12], or other binary rules that support gliders and glider-guns[13,14, 15, 26]. We can find large samples of complex rules by classifying rule-spaceautomatically according to the variability of input-entropy[31, 32, 38]. Withinthese samples a significant proportion are glider rules.We focus on 3-value k-totalistic glider CA on a 2D hexagonal lattice (andsome extension to 3D) with neighbourhoods of 6 or 7, including two well knownCA rules that have been studied in depth, the Beehive rule[2, 32, 33], and theSpiral rule[3, 34, 35]. The CA pulsing model is interesting in its own right, andmay also help to understanding and model oscillatory networks in biology. Weaddress the questions that arise about possible mechanisms, thought this paperis primarily concerned with presenting and documenting the phenomena.The paper is organised as follows: Section 2 describes CA and random wiring.Section 3 defines 3-value k-totalistic rules. Section 4 defines input-frequency andinput-entropy. Section 5 presents detailed pulsing case studies, including dif-ferent aspects of the waveform. Sections 7 — 9 examine the consequences offreeing one wire from localised neighborhoods, including 3D systems. In sec-tions 10 — 12 we discuss reaction-diffusion, asynchronous and noisy updating,and possible pulsing mechanisms. In sections 13 and 14 we discuss the impli-cations for bio-oscillations, ubiquitous at many time/size scales in biology, andfor modeling oscillatory behaviour in mammalian tissue such as the heart andcentral nervous system. Non-totalistic 3-value glider rules and 4-value k-totalistic glider rules, which are harderto find, will be examined in due course. The CA pulsing model has also been demonstrated for neighbourhoods of 4 and 5. CA and random wiring
55 00
2d cell=3,3=21 wiring=4,4 3,4 4,3 2,3 4,2 3,2 outwires=6 links:bi=108 self=0=0.0% k =6
55 00
2d cell=3,3=21 wiring=4,4 3,4 4,3 3,3 2,3 4,2 3,2 outwires=7 links:bi=108 self=36=14.3% k =7Figure 1: The (pseudo)-neighborhood template for hexagonal 2D CA, k =6 and k =7, with template cells numbered as in DDLab (for 3D see figure 16)
99 00
2d cell=5,5=55 wiring=8,8 1,5 0,8 0,1 4,8 6,2 outwires=6 links:bi=294 self=0=0.0% (a)
99 00
2d cell=5,5=55 wiring=5,1 4,6 8,0 0,1 0,7 9,9 2,3 outwires=9 links:bi=16 self=12=1.7% (b)
99 00
2d cell=5,5=55 wiring=5,2 7,9 1,3 7,0 8,4 2,7 3,1 outwires=6 links:bi=294 self=99=14.1% (c)Figure 2:
A hexagonal lattice 10 ×
10 showing k random cells (green) wired to thetarget cell, (a) and (b) via a pseudo-neighborhood template (yellow and red), and(c) directly. (a) For k =6 the target cell is not included in its neighborhood. (b)For k =7 the target cell (red) is included. For CA the actual neighborhood andpseudo-neighborhood are identical. (c) For k-totalistic rules and random wiring,strictly speaking, a pseudo-neighborhood template is not required. In classical CA, the pattern on the lattice updates (in discrete time-steps) aseach (target) cell synchronously updates its value according to the values inits local neighborhood template. In the general case the updating function is alookup-table (rule-table) of all v k possible neighborhood patterns, were v is thevalue-range and k is the neighborhood size.In this paper we focus mainly on 2D CA on a hexagonal lattice, with a lo-cal neighborhood “template” of k =6 and k =7 (figures 1 and 2), and also 3Din section 9. Boundary conditions are periodic (toroidal for 2D) — effectivelyno boundaries, but this is not significant in the CA pulsing model. The k =6template is shown in yellow surrounding the target cell, whereas the k =7 tem-plate includes the (red) target cell. Template cell numbers permit a completenon-totalistic rule-table to be assigned according to DDLab’s convention[37].To implement “random wiring”, as in Kauffman’s “Random Boolean Net-works” [20], for each target cell, we take k cells at random in the lattice and“wire” them to distinct cells in the pseudo-neighborhood template — “pseudo”because the actual template values are replaced by the values of the randomcells. Each target cell is assigned its own random wiring. The random wiringcan be biased in many ways in DDLab[37], one of which is to confine randomwiring within a local zone of arbitrary size (section 7). One or more wires can be“freed” from the zone, or from a CA neighborhood. Using DDLab, a single key Asynchronous and noisy updating is discussed in section 11.
We focus on 3-value k-totalistic rules for the following reasons: their rule-tablesare relatively short and thus tractable for displaying the input-frequency his-togram and its entropy (input-entropy); the dynamics are isotropic so closer tonature; and the availability of samples of glider rules. At present gilder rules v3k6 kcodeSize=28(hex) 0a0282816a0264(kcode-table:2-0)0022000220022001122200021210vfreq=11+4+13=2827: 6 0 0 -> 026: 5 1 0 -> 025: 5 0 1 -> 224: 4 2 0 -> 223: 4 1 1 -> 022: 4 0 2 -> 021: 3 3 0 -> 020: 3 2 1 -> 219: 3 1 2 -> 218: 3 0 3 -> 017: 2 4 0 -> 016: 2 3 1 -> 215: 2 2 2 -> 214: 2 1 3 -> 013: 2 0 4 -> 012: 1 5 0 -> 111: 1 4 1 -> 110: 1 3 2 -> 29: 1 2 3 -> 28: 1 1 4 -> 27: 1 0 5 -> 06: 0 6 0 -> 05: 0 5 1 -> 04: 0 4 2 -> 23: 0 3 3 -> 12: 0 2 4 -> 21: 0 1 5 -> 10: 0 0 6 -> 0\ - - - \\ 2 1 0 kcode (outputs)\ \\ totals of 2s, 1s, 0s\ in the neighborhood\kcode index (a) v k kcodeBeehive rule[32] v3k7 kcodeSize=36(hex) 020609a2982a68aa64(kcode-table:2-0)000200120021220221200222122022221210vfreq=18+6+12=3635: 7 0 0 -> 034: 6 1 0 -> 033: 6 0 1 -> 032: 5 2 0 -> 231: 5 1 1 -> 030: 5 0 2 -> 029: 4 3 0 -> 128: 4 2 1 -> 227: 4 1 2 -> 026: 4 0 3 -> 025: 3 4 0 -> 224: 3 3 1 -> 123: 3 2 2 -> 222: 3 1 3 -> 221: 3 0 4 -> 020: 2 5 0 -> 219: 2 4 1 -> 218: 2 3 2 -> 117: 2 2 3 -> 216: 2 1 4 -> 015: 2 0 5 -> 014: 1 6 0 -> 213: 1 5 1 -> 212: 1 4 2 -> 211: 1 3 3 -> 110: 1 2 4 -> 29: 1 1 5 -> 28: 1 0 6 -> 07: 0 7 0 -> 26: 0 6 1 -> 25: 0 5 2 -> 24: 0 4 3 -> 23: 0 3 4 -> 12: 0 2 5 -> 21: 0 1 6 -> 10: 0 0 7 -> 0 (b) v k kcodeSpiral rule[34] Table 1:
The kcode is a rule-table listing the output for every combination ofvalue totals in the neighborhood. For a system with 3 values (colors) the list isordered by the number of 2s, 1s, 0s, taken as a decimal number. The kcode isthen a string listing each output in descending order, from left to right, whichcan a be converted to hexadecimal for compactness. In DDLab these methods areimplemented automatically, for v ≤ and k ≤ . These examples show the kcodefor the Beehive rule and Spiral rule — their pulsing dynamics are examined below. S = ( v + k − / ( k ! × ( v − v k S =28, and for v k S =36, taking as examples the Beehiverule[32] and the Spiral rule[34]. The size of k-totalistic rule-space is v S . (cid:38) (a) ↓ (b) ↓ (c) Figure 3:
Dynamic graphics in DDLab show up pulsing in the v k Beehiverule[32] (hex 0a0282816a0264) on a 100 ×
100 hexagonal lattice with fully randomwiring. The period varies between 7 and 8 time-steps.(top) A typical sequence of the space-time patterns displaying pulsing densities, andrelated input-histograms, repeating on the 8th time-step. The horizontal bars rep-resent the lookup-frequency of 28 neighborhoods, as in figure 1(a). Below (arrowsshow time’s direction): (a) The input-histogram shown scrolling with time (z-axis).(b) Plotting the input-histograms values (x-axis) for successive time-steps (y-axis).(c) Input-entropy values (x-axis) plotted for successive time-steps (y-axis). The input-frequency and input-entropy
The input-frequency histogram tracks how frequently the different entries ina rule-table are actually looked up. This is usually averaged over a mov-ing window of w time-steps[31] to classify rules by the variability of input-entropy, but to track pulsing dynamics we take the measures over each time-step. The input-entropy is the Shannon entropy H of this input-frequencyhistogram. H , at time-step t , for one time-step ( w =1), is given by H t = − (cid:80) S − i =0 ( Q ti /n × log ( Q ti /n )), where Q ti is the lookup-frequency of neighbor-hood i at time t . S is the rule-table size, and n is the network size. Thenormalised Shannon entropy H N is a value between 0 and 1, H N = H t /log n ,which measures the heterogeneity of the histogram — henceforth “entropy” willrefer to H N . Figure 3 shows how space-time patterns, their input-frequency(histogram), and the input-entropy measures are tracked by dynamic graphicsin DDLab to show up pulsing, taking the v k We present six case studies of the CA pulsing model, based on glider rule samplesassembled previously[37], three for k =6 (figures 5-7), and three for k =7, (fig-ures 8-10), selected for a variety of waveform profiles. Surprisingly, wave-lengths( wl ) are very diverse, with average wl varying between 6 and 82 time-steps. Twowell documented rules are included, the v k v k ×
100 hexagonallattice. Random wiring is unconstrained (giving the “RW-waveform”) wherethe k inputs to each cell are independently assigned at random without bias.This makes the 2d geometry irrelevant — it is retained for convenience.The results are displayed graphically as follows: • (a) A typical snapshot before the wiring was randomised, of the CA withits emergent gliders, with 10 time-step green trails. • (b, c) Two snapshots after the wiring was randomised, showing the nowdisordered pattern (b) at its minimum and (c) maximum density of non-zero values (2=black, 1=red, 0=white). • (d, e) Space-time patterns showing evidence of pulsing, with cells coloredaccording to lookup instead of value, following the histogram colors infigure 3. (d) 2D space-time patterns scrolling diagonally, with the latesttime-step at the front, leaving a trail of time-steps behind. (e) A stretchof the space-times pattern transformed to 1D, scrolling vertically, with thepresent moment at the bottom, leaving a trail of time-steps behind. • (f1) The input-entropy plotted for each time-step, showing the pulsingwaveform. (f2) A stretched or magnified version of this plot, noting thewavelength wl and wave-height wh .6 (g) The entropy-density scatter plot — input-entropy (x-axis) against thenon-zero density (y-axis), plotted as blue dots, for about 33000 time-steps. • (h) The density return-map scatter plot — the density of each value at t (x-axis) against its density at t , plotted as colored dots (2=black, 1=red,0=green), for about 33000 time-steps.Case study experiments (confirmed for many other glider rules) give the fol-lowing general results: Any random initial state, (within reason ) will initiate atransient that rapidly converges on the waveform, which is impervious to reason-able noise. With fully random wiring (without bias), changing the actual wiringmakes no difference to the waveform, neither does re-randomising at each time-step. The scatter plots, both input-entropy and the density return-map, showunique signatures, which have the characteristics of chaotic strange attractors inthe context of deterministic discrete dynamical systems — sensitivity to initialconditions evolving towards a compact global attracting set, local instability butglobally stability. Varying the network size also preserves the waveform — thesignatures are diffused for small sizes, becoming more focused as the size of thenetwork increases (figure 4), and this would continue towards infinity. Reducingthe network size, however, increases the probability of reaching a uniform valueattractor, such as all zeros, where the system would stop.The “RW-waveform” results of these case studies will serve as a base ofcomparison with the other wiring biases investigated: CA with freed wires,localised random wiring (and with freed wires), and the equivalent in 3D. ×
50 100 ×
100 200 × v k “g26” rule, see figure 7(g).50 ×
50 100 ×
100 200 × v k “g1” Spiral rule, see figure 8(h). Figure 4:
Scatter plots (for about 33000 time-steps) become focused as the sizeof the nework increases, but the underlying signature is preserved. Initial states with some of the 3-values missing, or with very low/high density may notconverge on the waveform. a) (b) (c)Space-time pattern snapshots, (a) CA showing emergent gliders. Randomised wiring resultsin disordered patterns, (b) minimum density, and (c) maximum density.(d) (e) (f1)Space-time patterns illustrating density oscillations. (d) scrolling diagonally, the presentmoment is at the front leaving a trail of time-steps behind. (e) a 1d segment, scrollingvertically with the most recent time-step at the bottom. (f1) input-entropy oscillationswith time (y-axis).(f2) (g) (h)Time-plots of measures. (f2) input-entropy oscillations with time (y-axis, stretched) wl = 7 or 8 time-steps, wh ≈ . . (g) entropy-density scatter plot – input-entropy (x-axis)against the non-zero density (y-axis). (h) density return map scatter plot. Figure 5:
Pulsing dynamics for the v k “g2” Beehive rule, (hex) 0a0282816a0264,on a 100x100 hexagonal lattice, showing space-time patterns — snapshots, scrolling,and time-plots of measures. a) (b) (c)Space-time pattern snapshots, (a) CA showing emergent gliders. Randomised wiring resultsin disordered patterns, (b) minimum density, and (c) maximum density.(d) (e) (f1)Space-time patterns illustrating density oscillations. (d) scrolling diagonally, the presentmoment is at the front leaving a trail of time-steps behind. (e) a 1d segment, scrollingvertically with the most recent time-step at the bottom. (f1) input-entropy oscillationswith time (y-axis).(f2) (g) (h)Time-plots of measures. (f2) input-entropy oscillations with time (y-axis, stretched) wl
14 or 15 time-steps, wh ≈ . . (g) entropy-density scatter plot – input-entropy (x-axis)against the non-zero density (y-axis). (h) density return map scatter plot. Figure 6:
Pulsing dynamics for the v k “g39”rule, (hex) 0a184552558500(hex),on a 100x100 hexagonal lattice, showing space-time patterns — snapshots, scrolling,and time-plots of measures. a) (b) (c)Space-time pattern snapshots, (a) CA showing emergent gliders. Randomised wiring resultsin disordered patterns, (b) minimum density, and (c) maximum density.(d) (e) (f1)Space-time patterns illustrating density oscillations. (d) scrolling diagonally, the presentmoment is at the front leaving a trail of time-steps behind. (e) a 1d segment, scrollingvertically with the most recent time-step at the bottom. (f1) input-entropy oscillationswith time (y-axis).(f2) (g) (h)Time-plots of measures. (f2) input-entropy oscillations with time (y-axis, stretched) wl ≈ time-steps, wh ≈ . . (g) entropy-density scatter plot – input-entropy (x-axis)against the non-zero density (y-axis). (h) density return map scatter plot. Figure 7:
Pulsing dynamics for the v k “g26” rule, (hex) 1000a121960214, ona 100x100 hexagonal lattice, showing space-time patterns — snapshots, scrolling,and time-plots of measures. a) (b) (c)Space-time pattern snapshots. (a) CA showing emergent gliders. Randomising wiringresults in disordered patterns, (b) minimum density, and (c) maximum density.(d) (e) (f1)Space-time patterns illustrating density oscillations. (d) scrolling diagonally, the presentmoment is at the front leaving a trail of time-steps behind. (e) a 1d segment, scrollingvertically with the most recent time-step at the bottom. (f1) input-entropy oscillationswith time (y-axis).(f2) (g) (h)Time-plots of measures. (f2) input-entropy oscillations with time (y-axis, stretched) wl = 6 or 7 time-steps, wh ≈ . . (g) entropy-density scatter plot – input-entropy (x-axis)against the non-zero density (y-axis). (h) density return map scatter plot. Figure 8:
Pulsing dynamics for the v k “g1” Spiral rule, (hex)020609a2982a68aa64, on a 100x100 hexagonal lattice, showing space-time patterns— snapshots, scrolling, and time-plots of measures. a) (b) (c)Space-time pattern snapshots. (a) CA showing emergent gliders. Randomised wiringresults in disordered patterns, (b) minimum density, and (c) maximum density.(d) (e) (f1)Space-time patterns illustrating density oscillations. (d) scrolling diagonally, the presentmoment is at the front leaving a trail of time-steps behind. (e) a 1d segment, scrollingvertically with the most recent time-step at the bottom. (f1) input-entropy oscillationswith time (y-axis).(f2) (g) (h)Time-plots of measures. (f2) input-entropy oscillations with time (y-axis, stretched) wl ≈
21 time-steps, wh ≈ . . (g) entropy-density scatter plot – input-entropy (x-axis)against the non-zero density (y-axis). (h) density return map scatter plot. Figure 9:
Pulsing dynamics for the v k “g3” rule, (hex) 622984288a08086a94, ona 100x100 hexagonal lattice, showing space-time patterns — snapshots, scrolling,and time-plots of measures. a) (b) (c)Space-time pattern snapshots. (a) CA showing emergent gliders. Randomised wiringresults in disordered patterns, (b) minimum density, and (c) maximum density.(d) (e) (f1)Space-time patterns illustrating density oscillations. (d) scrolling diagonally, the presentmoment is at the front leaving a trail of time-steps behind. (e) a 1d segment, scrollingvertically with the most recent time-step at the bottom. (f1) input-entropy oscillationswith time (y-axis).(f2) (g) (h)Time-plots of measures. (f2) input-entropy oscillations with time (y-axis, stretched)diverse wl between 52 and 122 time-steps (average wl ≈ wh ≈ . . (g) entropy-density scatter plot – input-entropy (x-axis) against the non-zero density (y-axis). (h)density return map scatter plot. Figure 10:
Pulsing dynamics for the v k “g35” rule, (hex) 806a22a29a12182a84,on a 100x100 hexagonal lattice, showing space-time patterns — snapshots, scrolling,and time-plots of measures. Freeing one wire from CA neighborhoods (f2) wl ≈ v k “g2, beehive” (h)(f2) wl ≈
10 (g) v k “g39” (h)(f2) wl ≈
15 (g) v k “g26” (h) Figure 11:
Pulsing measures for 2D CA with one free wire, for the three v k casestudy rules in figures 5, 6 and 7. (f2) input-entropy/time plot, (g) entropy-densityscatter plot, (h) density return map scatter plot, for a 100x100 hexagonal lattice. If one wire is released from each neighborhood in the 2D CA, and freely con-nected anywhere in the lattice, glider dynamics is destroyed and we may beginto see pulsing. Freeing one wire results in significant pulsing in all three k =6rules in our case study (figure 11), and is also probable in other k =6 glider rules.For k =7, pulsing is less probable because a smaller proportion of the neighbr-hood is randomised — only one rule from the case study gave distinct pulsing14 f2) wl ≈
16 (g) v k “g3” (h) Figure 12:
Pulsing measures for 2D CA with one free wire for v k — only one casestudy rule, “g3” from figure 9 showed significant pulsing. (f2) input-entropy/timeplot, (g) entropy-density scatter plot, (h) density return map scatter plot, for a100x100 hexagonal lattice. (figure 12). If two wires are freed, pulsing is highly probable for both k =6 and k =7, and with more free wires pulsing properties approach the RW-waveform.The waveform is unaffected by re-randomising at each time-step.
2d cell=150,150=45150 wiring=141,157 143,150 149,145 147,142 157,159 154,145 outwires=6 links:self=0=0.0%
2d cell=20,20=820 wiring=10,29 13,25 28,20 29,10 28,27 11,13 outwires=6 links:bi=4794 self=0=0.0%
Figure 13: (Left) Random wiring confined within 20 ×
20 local zones within a300 ×
300 hexagonal lattice ( f = ≈ How does the dynamics play out if random wiring is confined within a local zonerelative to each target cell? — as in figure 13, which makes the 2D geometryof the lattice significant, whereas with fully random wiring the geometry losessignificance. Experiment shows that as the local zone diameter d is reducedrelative to the network diameter D — the reach of random wiring — overallpulsing, though still apparent, turns into patchy waves of density. At somethreshold (of the fraction f = d/D ) the stability and shape of the waveform willstart to deform relative to the RW-waveform, and eventually break down, aprocess that could be interpreted as a type of phase transition, though a properdescription will require further research and analysis. Preliminary results showthat the threshold f T is independent of network size, but varies according to15 k “g2” v k v k “g26” v k v k “g3” v k “g35” Figure 14:
Waves of density emerge when random wiring is localised within 20 × ×
300 hexagonal lattice (figure 13). Typical pattern snapshots areshown, with cells colored according to neighborhood lookup instead of value.(Top row) v k rules as in figures 5-7. (Bottom row) v k rules as in figures 8-10. Any initial state will set off similar dynamics. Overall entropy pulsing is stillapparent, and also patchy and spiral waves of density. the rule. For v k f T ≈ Freeing just one wire from the 20 ×
20 localised random zone in section 7, al-lowing it to connect anywhere, restores pulsing behaviour, but with a patchydistribution of values. Experiment confirms this applies to all the rules inthe pulsing case studies in section 5 –10. The waveforms are still recognis-able when compared to the RW-waveforms, including the entropy-density anddensity return-map scatter plot signatures. For example, the v k f2) wl ≈
21 (g) v k “g3” (h) v k “g3-min” v k “g3-max” Figure 15: ×
300 2D hexagonal lattice with random wiring confined within20 ×
20 local zones, but one wire freed, rule v k “g3” (from figure 9).(Top) Pulsing measures: (f2) input-entropy/time plot, (g) entropy-density scat-ter plot, (h) density return map scatter plot, with a strong similarity to the RW-waveform. (Bottom) Pulsing patterns at the extremes of input-entropy, with cellscolored according to lookup instead of value.
00 3d cell=1,1,1=13 wiring=1,1,2-1,2,1-2,1,1-0,1,1-1,0,1-1,1,0-outwires=6 links:bi=81 self=0=0.0% (a)
3d cell=22,22,22=45562 wiring=30,19,1-12,36,1-20,11,41-0,14,20-39,34,2-13,1,14-42,24,18-outwires=6 links:self=91124=14.3% (b)
3d cell=22,22,22=45562 wiring=24,20,24-21,21,20-20,20,21-28,8,43-20,22,23-21,23,24-21,24,21-outwires=6 links:self=91124=14.3% (c)Figure 16: (a) 3D neighborhoods, k =6 and k =7. (b) 3D 45 × ×
45 lattice withunrestrained random wiring. (c) Random wiring restrained in a 5 × × k =6 and k =7 CA (figure 16) were implemented in a 3D cubic lattice 45 × × × × (f2) wl ≈
21 (g) (h)(f2) wl ≈
20 (g) (h)
Figure 17: × ×
45 3D lattice with random wiring, rule v k “g3” — comparewith the RW-waveform in figure 9. Pulsing measures: (f2) input-entropy/timeplot, (g) entropy-density scatter plot, (h) density return map scatter plot. (Top)Unconstrained random wiring gives the same RW-waveform. (Bottom) Confinedwithin 5 × ×
10 k-totalistic rules as reaction-diffusion systems
An explanation of glider dynamics in k-totalistic rules can be based on Adamatsky’sreinterpretation of the k =6 Beehive rule[2], and the k =7 Spiral rule[34, 3], asdiscrete models of reaction-diffusion systems with inhibitor/activator reagents in18f2) wl ≈
21 (g) v k v k v k × ×
45 3D lattice with random wiring confined within 5 × × v k “g3” (compare with figures 17 and 9). (Top) Pulsing measures(f2) input-entropy/time plot, (g) entropy-density scatter plot, (h) density returnmap scatter plot — distorted compared to the RW-waveform. (Bottom) Pulsingpatterns at the extremes of input-entropy showing patchy waves of density — cellscolored according to lookup instead of value. a chemical medium. The three CA values are seen as: A=1 (Activator), I=2 (In-hibitor), and S=0 (Substrate). The three reagents perform a sort of non-linearfeedback dance, suppressing and catalysing each other at critical concentrations.The analysis accounts for the movement of a glider’s head and following tail,but could also apply to the randomly wired system seen as a neural networkwith three states: 1=(Activator, Firing), 2=(Refractory), 0=(Ready to Fire).When wiring is randomized, it seems that feedback becomes distributed, givingglobal pulsing instead of driving a glider.In glider CA, gliders and their interactions quickly dominate the dynamics,and thus the frequency of neighborhood lookup in the rule-table. The neighbor-hoods responsible for the background “domain” are the most frequent, followedby neighborhoods that drive gliders, other (stable) structures, and those in-vloved in collisions. The remaining neighborhoods rarely appear in an evolvedsystem and can be regarded as wild-cards in the rule-table — mutations of thesehave little or no effect[33, 35]. 19 Figure 19:
Lookup-histograms averaged over 100 time-steps, for k k ×
100 lattice. (Left) CA, (Right) Random Wiring,showing a correlation in neighborhooh frequency.
This is captured by the lookup-frequency histograms (figure 19) for the Bee-hive and Spiral rules[32, 34], averaged over 100 time-steps, where the CA his-togram highlights the background domain and glider dynamics, and the wild-cards by gaps or reduced values. The histogram for random wiring has a lesspronounced distribution, but there is a significant correlation with the CA his-togram, showing that the feedback between the three values is at play globally.Histograms for the other rules studied confirm these results.
11 Asynchronous and noisy updating
In the results so far, updating the next time-step has been deterministic, andsynchronous (in parallel) across the lattice — but what would be the effects ofnoise and asynchronicity? DDLab has a suite of options to introduce either orboth on-the-fly[37]. Two types of noise are implemented where each cell updateswith a given probability at each time-step — otherwise, in one alternative thecell stays the same, and in the other its value is assigned randomly. For asyn-chronicity, the most flexible method is “partial order” updating where a subsetof cells update (synchronously or sequentially), followed by the next subset —20 f2) (g) (h)Time-plots of measures. (f2) input-entropy oscillations with time (y-axis, stretched). (g)entropy-density scatter plot – input-entropy (x-axis) against the non-zero density (y-axis).(h) density return map scatter plot.
Figure 20:
Sequential updating in a random order, re-randomised at each time-step, showing pulsing measures taken at time-step intervals, which can be comparedwith figure 9. v k “g3” rule, (hex) 622984288a08086a94, on a 200 ×
200 hexagonallattice, wl = 11 or 12 time-steps. (a) 200 × ×
50, single-cell update
Figure 21:
Sequential updating in a random order within partial order updating,showing the entropy-density scatter plot. v k “g3” rule with a similar waveform asin figure 20. (a) Partial order limits: 1 to n , measures taken at time-step intervals.(b) Partial order limits: 1 to 1, measures taken at each cell-update. Starting at arandom initial state, pulsing completed only 3 cycles before measures fell to zero —this is expected because of the smaller network size, necessary because computationfor single-cell update is slow. then the “state” is the configuration after each updated subset. Lower and upperlimits predefine the size of each subset between 1 and network size n . At eachtime-step, a random size is set between these limits, and scattered randomlyto positions in the network — only those are updated. In sequential updating,each cell is updated in turn in some arbitrary order — then the “state” is theconfiguration when all n updates (or all cells in a partial order subset) are com-21lete. For n cells there are n ! possible sequential updating orders, but the usualmethod is to set a random order, re-randomised at each time-step. Using theseasynchronous and noisy updating methods, singly or in combination, it appearsthat the CA pulsing model continues to pulse whatever you throw at it.Sequential updating may seem biologically implausible because neurons donot wait for each one to fire in an orderly queue, but it avoids the critiqueof an artificial “synchronising mechanism” in synchronous models[18]. Withsequential updating the time-step becomes just a way of taking a look at thelattice at regular intervals (and taking measures). From this point of view, anypulsing must be a natural property of the rule, the wiring, and time seen as aseries of events. Amazingly, it turns out that pulsing continues when subjectto sequential updating (with or without constraints on duplication), thoughwith a reduced waveform . From this result, it is reasonable to conjecture that“natural pulsing” in the sequential case is also the driver (though stronger)when updating is synchronous.Figure 20 gives an example of sequential updating without duplication be-tween time-steps. Figure 21 shows two examples of sequential updating withinpartial order updating (with no constraint on duplication). Note that a partialorder size of exactly one results in completely arbitrary sequential single-cellupdates, with measures taken at each cell-update. The pulsing waveforms (forrule v k
12 Questions on the pulsing mechanism
The reaction-diffusing approach in section 10 is promising, and explains theneed for three (or more values) — two are not enough, but questions remain.Unravelling the CA glider mechanism itself, or predicting glider dynamics froma rule-table, are still unresolved questions in complex systems — answers wouldshed light on the underlying principles of self-organisation. The mechanism ofpulsing in the CA pulsing model is similarly unresolved — both phenomena areemergent. Gliders are mobile oscillating/repeating patterns in space and time,driven by feedback mechanisms within and between the neighborhood outputssurrounding the glider, and within the glider itself. Randomised wiring dispersesand synchronises these feedbacks over the whole network — pulsing must be aconsequence, where the 3-value densities in the disordered pattern fall into arepeating rhythm. The mobility aspect of well formed gliders are an essentialingredient, because experiment shows that dynamics showing up as “complex”in an automatic entropy-variability search[31, 32] but lacking gliders, do notpulse. These patterns include dynamic patches, blinkers, as well as mobileboundaries between ordered/disorderd domains. Further work will be requiredto define a well formed glider, and the significance of mobility to pulsing. Rules with a larger RW-waveform ( wl and wh ) from the pulsing case studies (section 5)continued to pulse with sequential updating, but the Beehive and Spiral rules with a smallRW-waveform did not.
13 Relevance to bio-oscillations
Pulsing — sustained periodic oscillations — are ubiquitous in many dynamicbio-cellular processes based on collective network behaviour, at a variety ofscales in both time and space, from cycles in gene expression to the rhythmof the beating heart. Some tentative models of bio-oscillations have been sug-gested: reaction diffusion , Hopfield networks, and attractors in discrete dy-namical networks[30, 37].The CA pulsing model described in this paper, where randomised wiring isapplied to 3-value k-totalistic CA with emergent glider dynamics, is arguablyrelevant to bio-oscillations, and may serve as a model that provides pointers to We note that pulsing from de-localising the connectivity in chemical excitable media hasbeen previously reported in the Belousov-Zabotinsky Reaction (BZR) though it is not clearthe significance was recognised at the time. The BZR in a complex chemical reaction-diffusionsystem (section 10) with more than 20 chemical reactions, time delays and the autocatalyticaccumulation of HBrO . Spirals in 2D and 3D gels are converted to whole system oscillationsin solution when stirred. Stirring presumably simulates the conversion of local to non-localconnectivity, re-randomised at each time-step as in Derrida’s quenched model[9]. In thischemical model pulsing frequency can be altered by temperature and concentration, andmaintained with a constant infusion of reagents[27]. • neurohormonal systems, • synchronised uterine contractions, • the Sinoatrial Node generating the heart rate, • the atria and ventricular chambers of the heart, • Central Pattern Generators (CPGs) of the brainstem and spinal cord,producing the following patterns of neural activity: – the sympathetic centre in the Rostral-Ventro-Lateral Medulla (RVLM)controlling sympathetic tone, the size of the vascular space, venousreturn and hence cardiac output and its distribution. – the pre-B¨otzinger cluster of interneurons in the ventral respiratorycentre of the medulla controlling the respiratory period. – the CPGs in the spinal cord underlying rhythmic motor behaviourssuch as walking, swimming, and feeding. • and the basic cortical building block the microcolumn[25] and hence per-haps the basic physiological building block of brain function.It is proposed that non-localised network connectivity combined with bio-logical processes similar to the glider rules described may have been favoured byevolution for the generation of robust biological oscillations due to the followingfunctional advantages: • the waveform (as defined in section 1) is dependant upon the rule of com-munication. • the waveform is independent of the exact wiring of the network, i.e. ran-dom within constraints (sections 6 – 9). • the waveform and its phase are robust to noise, perturbation and variabletransmission (section 11). • there is a phase transition between disorganised (absence of waveform) andorganised (presence of a waveform) behaviour, which occurs at a thresholdof network connectivity radius relative to the size of network (section 7). • if connectivity radius is above threshold, the waveform is robust to changesin the network size, and robustness is enhanced by increasing the radius.This built in redundancy affords physiological reserve.In theory, this results in randomly connected masses capable of robust os-cillatory behaviour in the presence of noise. The waveform can be modified bychanging the rule. Oscillatory behaviour can be turned off and on by alterationsin functional connectivity alone. The period of oscillation can be increased anddecreased (section 14.3). As a result there is potential to store information24etween weakly coupled robust controllable oscillators that is not present innon-robust oscillators in noisy systems.We find it significant that the above behaviour is emergent from a simplecomputational model with minimal conditions. The system requires 3 or morestates, a glider rule, non-local connectivity, a fixed number of connections, andto be thermodynamically open. No time delay for connection distance has beenincluded. Periodic boundary conditions are not required.For this model to be applicable to biological systems the following biologicalequivalents are required to exist within each system: • a biological unit with 3 (or possibly more) biological states. Traditionallyin excitable tissue these states are: Firing (F), Refractory (R), and Readyto Fire (RF) — (F.R.RF). • a biological process that has similarities to a rule with glider behaviourfor moving between these states. • a biological mechanism for non-local connectivity between the units. • and to provide variability in period and the ability to turn oscillations offand on: – a biological mechanism of speeding up and slowing down or evenhalting the biological process underlying the rule. – a method of altering biological connectivity, be it functional (shortterm) or structural (long term).We entertain the following questions/possibilities which will require furtherresearch: Do these biological equivalents exist within biological excitable tis-sues? And if they do, does the inheritance of the above properties minimise thestructural requirements of a system to fulfil its function? Or in other words cansophisticated behaviour be constructed from clusters of non-locally randomlyconnected glider rule system equivalents? These could be coupled in phase byexcitatory connections, coupled out of phase by inhibitory connections or evennon-locally coupled in a random way by a glider rule system.
14 Modeling bio-oscillations
CA glider rules are of interest in modeling excitable biological media as theypossess the following similarities: by definition both gliders and action poten-tials are patterns of state change that pass through a point in a medium whichafter its passage is left unchanged, they have a defined period and form, theycan be produced spontaneously and can annihilate each other. We propose pos-sible additional similarities observed with the non-localising of connectivity: theproduction of oscillations and the resultant emergant properties of this system(see section 13).Traditionally neurones and myocytes have 3 states - (F.R.RF). In muscle thefiring (F) state (contraction) results from increased intracellular Calcium con-centrations. Biological processes for moving between these states are Membrane25epolarisation/repolarisation (MD/R) by Voltage Gated Ion Channels (VGIC)and Calcium induced Calcium Release (CICR), and Calcium re-uptake fromintercellular stores by the Ryannodine Receptor and SERCA respectively[7]. Inneurones the firing (F) state is primarily associated with MD. MD can be spon-taneous or result from post synaptic integration of Post Synaptic Inhibitory andExcitatory Potentials (PSIP, PSEP). Presynaptic neurotransmitter release is aresult of increased intracellular Calcium in association with MD.This 3 state interpretation may place restrictions on CA rules representingbiological processes as by definition a cell cannot move between a Refractory andFiring state. An alternative is to equate each state to the Nernst potential foran ion. The Nernst Potential is the voltage a cell membrane will move towardsif membrane channels allowing conductance of that ion are opened. This resultsin 3 (or more) Voltage “Rails” in excitable tissue , so in the CA pulsing modela high density of the values 0, 1 and 2 would represent:0. the Resting Membrane Potential (RMP) or most VGIC in the closed state.1. a high density of open sodium VGIC or the N a + Rail ≈ +70 mV.2. a high density of open Calcium VGIC or the Ca Rail ≈ +120 mV.Cells in the model, dependant on scale, can represent the density of ion chan-nel opening in a membrane, or the membrane potential at membrane, cellularor grid levels.Examination of the rule-table and its input-frequency histogram indicatethe network trends through the rule-table in a series of steps as a result ofdensity fluctuations before returning close to its starting configuration at period(section 3, figure3). Experimentally, pulsing still occurs when random noise isintroduced to the deterministic system (section 11). The phase of pulsing isunaltered by randomising wiring at each time-step. This indicates that if noiseintroduced to the system is less than the density difference between time-steps,the system will continue to pulse. The emergent pulsing behaviour is essentiallyindependent from, and insensitive to, initial conditions and a degree of noise.How this model relates to biological processes such as robust oscillationsproduced by positive and negative feedback and a time delay[6], and otherquestions from section 12, should be further investigated. Below we suggest someoscillatory physiological systems where the CA pulsing model and its inherentproperties, because of its diversity of waveforms, might be usefully applied. Braxton Hicks contractions transition to synchronised uterine contractions calledlabour. It is known that labour and pre-term labour is associated with increasedGap Junction density. Gap Junctions electrically couple adjacent cells at ran-dom points in the cell membrane. Sophisticated computer models of the uterus Additional states can be introduced for additional ion channels. The K + ≈ -60 mV Railcould be differentiated from the RMP and in neuronal systems a Cl − Rail at ∼ -80 mV couldbe introduced. The heart could be acting as a possible instance of the CA pulsing model oper-ating in a real biological system. All myocardial cells in the atria, ventricles, andconducting system, have the potential to periodically fire by CICR, spontaneousmembrane depolarisation, or global membrane depolarisation, producing mor-phologically identifiable etopic beats. This indicates these systems over multiplescales have similar innate and entrained periodicities and confer the system con-siderable robustness. The heart is driven primarily by an anatomically poorlydefined group of “pacemaker” cells called the SinoAtrial Node (SAN). Thesecells have no Sodium VGIC, which could be thought of as a modified rule.Connectivity is effectively non-localised within constraints (section 7) acrossmultiple levels as follows: • the T-Tubular network non-locally connects the Cell Membrane to theSarcoplasmic Reticulum (SR)[17]. • the Gap Junction network connects the cytoplasm of adjacent myocytes.Connectivity varies with Connexin pore size[8]. • Myocytes are arranged in a non-grid like way as they themselves are ofdiffering length. There are a large number of other cells present[21]. • the ventricle requires an additional non-localised network to synchroniseits greater mass called the Purkinje system. It consists of longer myocyteswith less resistant, faster Gap Junctions (section 8).Arrhythmias or the breakdown of the normal heart rhythm can be categorisedas disorders of the rule, the absolute number of myocytes or connectivity, asfollows: • rule changes include channelopathies, acute ischaemia and severe elec-trolyte abnormality. The Vaughan-Williams classification classifies anti-arrhythmic medications by alterations to the rule. • loss of SAN cell mass with age typically results in alterations in heartrate variability (section 5) before pacemaker failure. Larger Atrial andVentricular chambers are more difficult to synchronise, increasing the riskof fibrillation and flutter. Rotors can be seen in non-pulmonary vein AtrialFibrillation (section 7). 27 loss of myocyte connectivity (scarring, fatty and fibrotic deposits, connexinchanges, inflammatory mediators, alterations in gene expression etc[4]) in-crease the risk of arrhythmias and sinus node disease. Peptides enhancinggap junction function and myocardial cell communication are currentlyunder investigation as a new class of anti-arrhythmic drugs[8]. It is worth contrasting the steady rhythm of the CA pulsing model, and theperiod of the underlying biological oscillation it may represent. The frequency ofthe bio-oscillation can change without changing the underlying rule equivalentby altering the absolute value of the time-step and hence period. From theCA point of view there would still be the same number of time-steps as in arepeating waveform. This is not modelled at present but would become relevantwhen modeling two or more biological processes with non identical rate change.Altering the speed of a biological process is subject to regional variance. Byway of example, release of Acetyl Choline (ACh) from the Vagus nerve on theSAN opens GPCR Coupled Potassium leak channels, slowing the rate of spon-taneous depolarisation towards VGIC threshold. It is unlikely the concentrationof ACh and the effect of Potassium conductance in every myocyte in the SANis identical, yet the heart rate slows without becoming disorganised. Perhapsnon-localisation of connectivity, and the synchronisation it affords, validates theextension of process change at the channel or membrane level to the behaviourof an excitable tissue mass as a whole.
The Central Nervous System is capable of acquiring new patterns, and repro-ducing them in either the long term (for example movement/memory) or inthe short term (working memory). Neurons are organised into functional units(CPG, brainstem nuclei, Microcolumns) by local connectivity (dendrites, shortaxons, gap junctions and synapses) with long range connectivity between units(long axons). It is estimated we each have ≈ × microcolumms comprising ≈
100 neurones each.Functional connectivity is determined by neurotransmitter synaptic and den-dritic release and post synaptic integration of Inhibitory and Excitatory PostSynaptic Potentials (IPSP and EPSP). Regional functional connectivity can bealtered by the neuromodulators via GPCR. Oscillatory behaviour is thought toemerge as IPSP’s are of longer duration than EPSP’s[11]. However non-localconnectivity may be a key component of the “fast” component of a fast/slowbiological system and the mean field model. Connectivity effects on these unitsare well modelled in DDLab by a 3D system with restrained random wiring withreleased wires (section 9). 28
Central Pattern Generators (CPGs) are neuronal clusters that produce: • Permanent Oscillations: The pre-Botzinger cluster or respiratory pace-maker can produce slow breathing, sniffing and gasping patterns depen-dant on input. The corresponding rhythm is projected throughout thecortex by noadrenergic neurones. • Driven Oscillations: Oscilations of the sympathetic centre in the RVLMare driven (at a shorter period than its own) by pulses of inhibition fromthe arterial waveform, (ie. carotid pressure sensors via a GABA-A in-terneuron) which then rebounds to its set point. Sympathetic nervoussystem abnormalities, both in set point and integration, are crucial in thedevelopment of cardiovascular disorders such as Heart Failure, EssentialHypertension and Postural Syncope[10]. • Controlled Oscillations: Locomotion CPGs in the spinal cord are turnedon and off by Neuromodulation. • Coupled Oscillations: Patterns of movement are stored as associationsbetween GPG’s and their respective muscle groups. In a similar fashionit may be possible to store patterns or information between associatedmicrocolumns.The evolution of rhythmic behaviours in the invertebrate and simple vertebratereveals repeated building blocks such as two mutually inhibitory half centreoscillators[19]. However the mammalian CPG is essentially a black box[16] dueto the vast numbers of neurones and their associations, and the difficulty ofobtaining simultaneous readings of their electrical activity. Without models thesituation may remain that way for some time.The CA pulsing model demonstrates significant biological emergent proper-ties: sustained rhythmic oscillations, a threshold effect, redundancy and robust-ness to noise. Thus the question arises whether the pre-Botzinger cluster, thesympathetic centre in the RVLM, and even motor CPGs need to be anythingother than masses of pseudo randomly connected neurones with associationsmapped to anatomical or physiological features, or other CPGs.
15 Discussion
To explain the pulsing mechanism is as hard as explaining the mechanismwhereby a CA rule acting on regular CA wiring is able to generate glider dynam-ics. Although its possible to discriminate between the extremes of ordered anddisordered dynamics from the rule-table, by the λ and Z parameters[22, 30, 31],the link between the CA rule and glider dynamics is still an open question, goingto the heart of the underlying principles of self-organisation. The mechanism ofthe glider and glider-gun, with its many delicate feedback loops, is also difficultto untangle. 29hat seems to be apparent is that for 3-value glider CA (but not binary,2-value) fully random wiring makes pulsing inevitable. Pulsing could be re-garded as temporal order emerging from the disordered patterns driven by therandomly connected CA. Starting with localised wiring, there is some sort ofphase transition to enhanced pulsing strength depending on the degree andreach of the random connections.The diversity of pulsing waveforms in glider CA with random wiring mayprovide models and insights into bio-oscillations in nature. Many attributes ofthe CA pulsing model are reflected in oscillatory behaviour in mammalian tissuesuch as the heart and central nervous system. The model provides a classifi-cation system for oscillations in biological systems, their formation and theirbreakdown according to the (biological) rule, network size, and connectivityrelative to threshold.In this paper we have introduced and documented the pulsing phenomenaand listed the issues that require explanation. Further systematic research andexperiment is required to properly investigate the range and scope of pulsing,its mathematical and logical properties, the mechanisms that drive it, and itsbiological significance.
16 Acknowledgements
Experiments and figures were made with DDLab ( ) —where the rules and methods are available, so repeatable[37].Thanks to Inman Harvey for conversations regarding asynchronous updat-ing, to Terry Bossomaier for exchanges regarding phase transitions, and toPaul Burt and Muayad Alasady for comments regarding bio-oscillations.
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