The cellular automaton pulsing model, experiments with DDLab
TThe cellular automaton pulsing model,experiments with DDLab ∗ Andrew Wuensche † Discrete Dynamics Lab.
Edward Coxon ‡ Dept. of Anaethesia and Pain Medicine,The Canberra Hospital, ACT, Australia.
Abstract
The cellular automaton (CA) pulsing model[1] described the surprisingphenomenon of spontaneous, sustained and robust rhythmic oscillations,pulsing dynamics, when random wiring is applied to a 2D “glider” rulerunning in a 3-value totalistic CA. Case studies, pulsing measures, possiblemechanisms, and implications for oscillatory networks in biology werepresented. In this paper we summarise the results, extend the entropy-density and density-return map plots to include a linked history, look attotalistic glider rules with neighborhoods of 3, 4 and 5, as well as 6 and7 studied previously, introduce methods to automatically recognise thewavelength, and extend results for randomly asynchronous updating. Weshow how the model is implemented in DDLab to validate results, outputdata, and allow experiments and research by others. keywords: cellular automata, glider dynamics, random wiring, pulsing,bio-oscillations, emergence, chaos, complexity, strange attractor, heartbeat
The cellular automaton pulsing model (the CAP model)[1] is a 2D cellular au-tomaton (CA) subject to a 3-value k -totalistic “glider” rule, where the localwiring is randomised. Pulsing dynamics, sustained rhythmic oscillations of den-sity and entropy measures, emerge spontaneously and almost inevitably. Thecharacteristic wave-forms are robust and depend on the specific glider rule ap-plied. If the extent or reach of random wiring is reduced, pulsing will eventuallybreak down, suggesting a threshold and phase transition. ∗ Presented at Summer Solstice 2018 Conference on Discrete Models of Complex Systems. † [email protected], ‡ [email protected] a r X i v : . [ n li n . C G ] N ov he CAP model is a significant phenomena in its own right, posing ques-tions in CA theory on the mechanisms of glider dynamics, the mechanisms ofpulsing, and how the two are related. It is also significant in the context of bio-oscillations ubiquitous at many time/size scales in biology. Currently there isno satisfactory theory to explain essential oscillations in various animal organs,for example heart beat, uterine contractions in childbirth, and various rhythmicbehaviours such as breathing controlled by the central nervous system. TheCAP model provides a possible oscillatory bio-mechanism based on long rangesignalling between cells/neurons in excitable tissue, following the classical threestate dynamic, Firing, Refractory, and Ready-to-fire, and subject to “gliderrule” equivalent logic.These concepts, ideas and results were defined and documented in [1]. Inthis paper we present a summary, extend the entropy-density and density-returnplots to include a linked history, and look at glider rules with smaller neighbor-hoods k , of 3, 4 and 5, as well as 6 and 7 studied previously. We introducemethods to automatically recognise and measure wave-length/wave-height andoutput the data. We extend results for randomly asynchronous updating. Weshow how the model is implemented in DDLab to allow validation of results andfurther experiments and research by others, and include pre-assembled collec-tions of glider rules.
99 00
2d cell=5,5=55 wiring=6,6 4,5 6,4 outwires=3 links:bi=0 self=0=0.0% (a) k =3
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2d cell=5,5=55 wiring=6,6 5,5 4,5 6,4 outwires=4 links:bi=0 self=100=25.0%
32 10 (b) k =4t
99 00
2d cell=5,5=55 wiring=5,6 6,5 4,5 5,4 outwires=4 links:bi=200 self=0=0.0%
32 10 (c) k =4s
99 00
2d cell=5,5=55 wiring=5,6 6,5 5,5 4,5 5,4 outwires=5 links:bi=200 self=100=20.0%
43 2 10 (d) k =5
99 00
2d cell=5,5=55 wiring=6,6 5,6 6,5 4,5 6,4 5,4 outwires=6 links:bi=300 self=0=0.0% (e) k =6
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2d cell=5,5=55 wiring=6,6 5,6 6,5 5,5 4,5 6,4 5,4 outwires=7 links:bi=300 self=100=14.3% (f) k =7 Figure 1:
2D neighborhood templates ( k =3 to 7) as defined (and numbered) inDDLab, setting the lattice geometry, both hexagonal and square. The target cell iscentral even if not part of the template. The lattice geometry of a 2D CA depends on its neighbourhood template, andwe present templates relevant to this paper in figure 1. Collections of gliderrules have been assembled based on these templates.Each cell in the lattice updates its value synchronously according to a ho-mogeneous 3-value k -totalistic rule. This determines the dynamics which can beseen as successive pattern images in the same way as a series of still images makea movie. Most rules result in disorder, but we are interested in complex rulescharacterised by identifiable mobile features, in particular “gliders” or mobileparticles consisting of compact cell-value assemblies moving through the latticewith a given velocity, comprising a head and tail, and interacting by collisionswith other gliders or stable particles as in the examples in figure 2. Although synchronous updating is a necessary condition for gliders to emerge, pulsing inthe CAP model persists for asynchronous and noisy updating[1]. a) v3k3x1.vco, g1(hex)00a864 (b) v3k4t1.vco, g1(hex)2a945900 (c) v3k4x1.vco(hex)2282a1a4(d) v3k5x1.vco, g1(hex)004a8a2a8254xx (e) v3k6n6.vco, g16(hex)01059059560040xx (f) v3k7w1.vco, g1(hex)020609a2982a68aa64The Spiral rule[2] Figure 2:
Examples of glider dynamics for 2D neighborhood templates ( k =3 to 7)in figure 1. Cell values: 0=white, 1=red, 2=black. Green trails of 5 time-stepsindicate glider velocity. Examples b, c, e, and f include glider-guns. The rules canbe loaded in DDLab by their filenames, in hexadecimal, or from the rule collectionsindex g(x). In DDLab, collections of glider rules are provided, extracted from automaticsamples of complex rules — not all complex rules support gliders and pulsing.The older collections for k = 6 and 7 relating to [1] include complex rules as wellas pulsing rules, whereas for the k = 3, 4, and 5 collections, only pulsing ruleshave been included. For k =3, pulsing results from mobile intersecting linearstructures, rather than classic gliders which are less frequent.While assembling these collections we can confirm that gliders almost in-evitably imply pulsing, and their absence imply non-pulsing. We should howevernote that we have observed a few very rare exceptions.3
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2d cell=20,20=820 wiring=39,33 0,27 23,7 8,0 12,36 11,14 5,12 outwires=3 links:bi=29 self=7=0.1% (a) unrestricted
2d cell=20,20=820 wiring=17,20 17,15 24,24 24,15 21,22 23,20 15,24 outwires=11 links:bi=297 self=119=1.1% (b) confined in a local zone
2d cell=20,20=820 wiring=24,19 36,1 17,22 24,21 18,24 19,16 17,15 outwires=6 links:bi=222 self=100=0.9% (c) confined but one wire freed
2d cell=20,20=820 wiring=37,36 19,21 21,20 20,20 19,20 20,19 19,19 outwires=6 links:bi=3327 self=1491=13.3% (d) CA but one wire freed
Figure 3:
Examples of random wiring in a 40 ×
40 2D lattice. A bigger latticewould normally be required for robust pulsing, to avoid reaching a uniform valueattractor[3].
Unrestricted, unbiased, random wiring follows the same connection approach asKauffmans’s “Random Boolean Networks”[4]. For each target cell, we take k cells at random anywhere in the lattice and “wire” them to distinct cells in thepseudo-neighborhood template — “pseudo” because the actual template valuesare replaced by the values of the random cells, as in figure 3(a). Each targetcell is assigned its own random wiring. We have also studied random wiringwith three types of bias resulting in degrees of degraded pulsing and waveformsignatures[1]. A summary of the consequences relating to figure 3(a,b,c,d) arelisted below,(a) Unbiased random wiring gives the strongest and most robust pulsing in theCAP model. 4b) If random wiring is confined within a local zone, as the size of the zone isreduced, at some stage global pulsing will break down into mobile patchesof density which may take the form of spiral waves. The threshold prop-erties of this phase transition requires further investigation. The waveformsignatures are recognisable as they degrade.(c) If one wire is “freed” from a small local zone, robust pulsing with a char-acteristic waveform is partially restored, and this is reinforced as a higherproportion of wires are freed.(d) Starting with a conventional CA, freeing one or more wires randomly fromeach neighborhood template results in the onset of pulsing by degrees, alsoconforming to a characteristic waveform.In these experiments, re-randomising the wiring at each time-step, as inDerrida’s annealed model[5], makes no significant difference to the waveform.A single key press in DDLab enables switching between CA and any type ofpreset random wiring, or between stable random wiring and re-randomising thewiring at each time-step while maintaining the preset bias. k -totalistic rules? We restrict our investigation to the subset of 3-value k -totalistic rules for thefollowing reasons: • The discovery of pulsing in the CAP model, and that no pulsing is evidentin an equivalent 2-value system. • Compared to a general CA, the k -totalistic rule-table is relatively short[10, • The dynamics are isotropic so closer to nature — the same output forneighborhood template rotation or reflection, though k -totalistic rules arerestricted beyond isotropy giving smaller rule-spaces than just isotropicrules. • The rules can be reinterpreted as reaction-diffusion systems with inhibitor-activator reagents in a chemical medium[6, 7, 2], where the three CA valuesare seen as: Activator, Inhibitor, and Substrate. • The availability of short-lists of glider rules, extracted from large sam-ples of complex rules that are found (and sorted automatically) by thevariability of input-entropy[8, 9, 10]. • The CAP model can be appled to bio-oscillations in excitable tissue ac-cording to classical 3-state neuronal dynamics: Firing, Refractory, andReady to Fire. 5
Definition of 3-value k -totalistic rules The properties and definition of 3-value k -totalistic rules are summarised asfollows: • The target cell at time step t depends on the combination of k totals, orfrequencies, of the values in the neighborhood template at t -1. • Each combination of totals make up the rule-table (named “kcode”), forexample the 3-value ( v =3) k =5 rule v3k5x1.vco in figure 2(d), In DDLab, kcode can be expressed in hexadecimal for compactness, inthis case 004a8a2a8254, also shown in figure 2(d). • kcode size S = ( v + k − / ( k ! × ( v − v S . A general rule-table (rcode) has v k entries increasingexponentially; rcode-space= v v k . For v =3 and k =3 to 7, the size of thekcode and rcode strings are as follows, k Figure 4:
Dynamic graphics in DDLab show up pulsing in the v k Spiral rule[2]from figure 2(f), with a period of 7 time-steps. Each 100 ×
100 pattern in a typicalcycle is shown above its input-histogram, where horizontal bars represent the lookup-frequency of 36 neighborhoods, (all-2s at the top) for each corresponding time-step. (a) (b) (c)
Figure 5:
For the v k Spiral rule from figure 2(f), (a) Input-entropy oscillationswith time (y-axis, stretched), wl ≈ time-steps, wh ≈ . . Left edge: superim-posed histogram values plots. (b) The entropy-density scatter plot — input-entropy(y-axis) against the non-zero density (x-axis), for about 33000 time-steps. (c) Thesame plot for just a few pulsing cycles, but linking successive dots giving a time-history. (a) (b) (c) Figure 6:
For the v k Spiral rule from figure 2(f), (a) Value-density oscillationswith time (y-axis, stretched). 0=green, 1=brown, 2=black. (b) The density return-map scatter plot — the density of each value at t (x-axis) against its density at t ,plotted as colored dots as above. for about 33000 time-steps. (c) The same plotfor just a few pulsing cycles, but linking successive dots giving a time-history. andplotted with its characteristic wavelength ( wl ), wave-height ( wh , twice am-plitude), and waveform (its shape or phase), which in turn can generate anentropy-density scatter plot[8] (fig 5). From space-time patterns, the density orproportion of each value, (0, 1, 2) if v =3, can be plotted, and this can generatea density return-map scatter plot[10] (fig 6). The scatter plots have the charac-teristics of chaotic strange attractors, and successive dots can be connected tocreate a linked history — this option is much faster to produce the character-istic plot because just a few time-step are needed. The wl and wh data can berecognised and output automatically.We will use the term “waveform” to sum up these pulsing measures. Eachglider rule in the CAP model maintains its distinctive waveform signature, re-flecting the distinctive glider dynamics. It was shown in [1] that the underlyingwaveform signature is independent of the network size n , becoming more fo-cused as n increases towards infinity, but reducing n makes reaching a uniformattractor[3] more likely, where the system would freeze.Waveform measures and plots are usually averaged over a moving window of w time-steps, w =10 to classify rules by the variability of input-entropy[8], butto observe pulsing dynamics most effectively we take the measures over eachtime-step where w =1. However, when measuring the wave-length wl automati-cally, w ≥ k =7 Spiral rule[2] on a 100 ×
100 lattice, with w =1. A new method is available in DDLab to automatically recognise and measurethe wave-length ( wl ) and wave-height ( wh ) of entropy oscillations in the CAPmodel. The output appears in the terminal. The data can be activated whilethe density-entropy plot is active. The algorithm is effective for well developedsteady (but possibly variable) entropy oscillations — the examples in figure 7 re-late to the rules in figs 9, 10, and 12. Entropy oscillations with jagged stretchesor transient min/max values in the plot profile, as in figures 8 and 12(a), candisturb the wave-length ( wl ) measures, but this can be smoothed out by mak-ing the time-step window w ≥ wl ), though ( wh ) would bereduced. Figure 8 gives an example of variable wl , and with jagged stretchesresolved by making w =20. Section 10.2 includes step-by-step instructions forthe method. The input-entropy is the Shannon entropy H of the input-histogram. For one time-step, H t = − (cid:80) S − i =0 (cid:0) Q ti /n × log (cid:0) Q ti /n (cid:1)(cid:1) , where Q ti is the lookup-frequency of neighborhood i attime t . S is the rule-table size, and n is the network size. The normalised Shannon entropy H N is a value between 0 and 1, H N = H t /log n , which measures the heterogeneity of thehistogram — “entropy” in this paper refers to H N . ( a) w =1, fig 9 v3k4 kcodeSize=15(hex)2a945900 w=1min=107 wl=6 wh==0.294max=110 wl=7min=114 wl=7 wh==0.332max=117 wl=7min=121 wl=7 wh==0.270max=123 wl=6min=127 wl=6 wh==0.355max=129 wl=6 ( b) w =1, fig 10 v3k5 kcodeSize=20(hex)004a8a2a8254 w=2min=116 wl=11 wh==0.428max=118 wl=11min=126 wl=10 wh==0.420max=129 wl=11min=138 wl=12 wh==0.457max=141 wl=12min=148 wl=10 wh==0.442max=151 wl=10 ( c) w =2, fig 12Figure 7: Wave-length ( wl ) and wave-height ( wh ) examples showing data for atypical sequence of 4 pulsing cycles. Data is output continuously in the terminal,and average values so far if interrupted. The rules, shown at the top, relate towaveform figures indicated. The algorithm identifies the time-step at the minimumand maximum values of each oscillation to calculate ( wl ) and ( wh ). The size of thetime-step window w is shown; usually w =1, but for example (c) w =2 to smoothout a jagged stretch at the maximum part of the plot profile in figure 12(a).( a) w =1 ( b) w =20 v3k7 kcodeSize=35(hex)806a22a29a12182a84 w=20min=691 wl=101 wh==0.206max=711 wl=101min=781 wl=90 wh==0.202max=800 wl=89min=893 wl=112 wh==0.210max=913 wl=113min=969 wl=76 wh==0.199max=989 wl=76min=1059 wl=90 wh==0.211max=1078 wl=89min=1133 wl=74 wh==0.211max=1152 wl=74min=1195 wl=62 wh==0.193max=1214 wl=62min=1258 wl=63 wh==0.203max=1278 wl=64min=1349 wl=91 wh==0.195max=1369 wl=91min=1427 wl=78 wh==0.199max=1446 wl=77min=1482 wl=55 wh==0.198av-wl=79.14, av-wh=0.201, sample=50 ( c) w =20 Figure 8:
Wave-length ( wl ) for rule v k g35 in [1], with steady but variable wl oscillations, and jagged stretches on the downslope of the entropy plot profile (a)can give false min/max results, but this is fixed by increasing the time-step window( w =20) to smooth the plot (b). (c) shows typical data, with wl between 62 and113 time-steps, though the actual range is slightly greater. The last line showsaverage values. CAP model plots for k =3, 4, 5, and 6 The CAP model input-entropy and value-density plots, as in figs 5 and 6, for the v k ×
100 lattice, are shown here for therules in section 2, for k =3, k =4 (triangular and orthogonal), k =5, and k =6. Foreach rule, four plots (a, b, c, d) described below, are shown in figs 9 to 13. Therules can be loaded in DDLab in various ways including by their filenames orin hexadecimal, but to explore the range of pulsing behaviours most effectively,from the rule collections index g(x).(a) Input-entropy oscillations with time (y-axis, stretched). Left edge: super-imposed histogram values plots.(b) The entropy-density scatter plot — input-entropy (y-axis) against the non-zero density (x-axis), for just a few pulsing cycles, and linking successivedots giving a time-history.(c) Value-density oscillations with time (y-axis, stretched). 0=green, 1=brown,2=black.(d) The density return-map scatter plot — the density of each value at t (x-axis) against its density at t , plotted as colored dots for just a few pulsingcycles, and linking successive dots giving a time-history. (a) (b) (c) (d) Figure 9: k3 CAP plots, v3k3x1.vco, (hex)00a864, g(1). (a) (b) (c) (d)
Figure 10: k4 (triangular) CAP plots, v3k4t1.vco, (hex)2a945900, g(1). a) (b) (c) (d) Figure 11: k4 (orthogonal) CAP plots, v3k4x1.vco, (hex)2282a1a4 (a) (b) (c) (d)
Figure 12: k5 CAP plots, v3k5x1.vco, (hex)004a8a2a8254, g(1) (a) (b) (c) (d)
Figure 13: k6 CAP plot, v3k6n6.vco, (hex)01059059560040, g(16)
Pulsing in the CAP model subject to asynchronous and noisy updating turnsout to be robust[1], but perhaps the most intriguing and unexpected result isthat pulsing continues with a recognisable waveform when random singe cells areupdated one at a time. Experiments for k =6 and k =7 glider rules Figures 14)confirm these results, where data is plotted for each cell update so experimentstake 10000 times longer (for 100 × For other rules and k values, the results are still under investigation. a) v k g3 wl ≈ wh ≈ v k g35 wl , wh see fig 8 Figure 14:
Entropy-density plots for sequential updating one cell at a time atrandom positions[1] result in pulsing. The network is 100 × n =10000 suchupdates are required to approximate one synchronous time-step. (a) v k g3 (seefig 16), and (b) v k g35 (see fig 8). Although the CAP is a computer simulation, the fact that the pulsing wave-form is preserved for randomly sequential singe cells are updating is significantin the sence that the CPU timer can be ruled out as an external time-keeper.
10 Experiments with DDLab
Using the DDLab software[11], the results presented in [1] and in this papercan be reproduced, and many other rules and aspects of pulsing dynamics in-vestigated. Pre-assembled collections of glider rules are available, and can beactivated on-the-fly (key g ) while space-time patterns are active, the wiringcan be toggled between CA and random (key ), and the dynamics observed,measured and recorded with other keys and interactive functions. The oldercollections for k = 6 and 7 relating to [1] include complex rules as well as pulsingrules. For the k = 3, 4, and 5 collections, only pulsing rules have been included— figure 15 shows an overlay of all 20 entropy-density plots for each of theserule collections.We summarise below the steps in DDLab to run the experiments, referringto chapters and sections (denoted by , from a terminal in the directory ddlab/ddfiles , enter ../ddlabz07 -w & to start in a white screen. Download the latest compiled version of DDLab (Nov 2018 or later) for Linux or Macfrom to a directory called ddlab , and the extra files in dd_extra.tar.gz to asubdirectory called ddlab/ddfiles (directory names are arbitrary). readme files and
Makefiles provided. =3 k =4 k =5 Figure 15:
For k = 3, 4, and 5, 20 overlaid entropy-density plots for just a fewpulsing cycles each, and linking successive dots giving a time-history. Read
Return to step forward, q to backtrack or interrupt. Return withoutinput selects a default.
For CAP pulsing experiments, we will set up a 100 ×
100 2D network where v =3,and k =7 (for example) and totalistic rules ( t for TFO-mode (Totalistic Forward Only, Value range prompt enter ( return until a top-right WIRING prompt window appears ( for hexagonal 2d ( for square/orthogonal).(4) At the next top-right prompt to set the 2d size ( for both i and j .(5) At the next top-right prompt, Neighborhood size k: ( .(6) At the next top-right
2d network ... wiring prompt ( to display a 2D wiring graphic bottom left, showing CA neighborhoods,together with a top right reminder ( b for a 2d block ( a to outlinethe “block” as the whole network ( r for random,which shows links for the highlighted cell as in figure 3(a). Click otherlocations to show the wiring for other cells.(8) Enter return until the top-right prompt revise from: ( e ,then at the entropy/density: prompt ( e again.(9) At the next top-right prompt, enter d to skip further special options andstart the 2d space-time patterns iterating in the top-left of the screen fora random rule, with the input-entropy plot (averaged for a moving windowof 10 time-steps) and the input-histogram alongside. The rule was set atrandom so oscillations are unlikely. To revise any of the above, backtrackwith q . 13igure 16: The DDLab screen showing 2D space-time patterns, 100 × v k g3 (hex)622984288a08086a94. While 2D space-time patterns are running, on-the-fly key-press options may beactivated/toggled ( q , which gives a top-right promptwindow with further options ( , key-press g ( q to backtrack to thepause prompt ( • enter G for a top-right prompt showing the number of rules in thecollection ( In the k =6 and k =7 collections, not all rules are glider rules, so not all will pulse. enter , or any valid number, to select the rule (not yet activated). • once space-time patterns resume, key-press g to activate the rule. • For the next rule enter g , eventually cycling back to rule 1.(3) Waveform output is most pronounced when set to each single time-step, notto the default 10 time-step moving average. To change this, key-press G and at a top-right prompt ( .(4) Important on-the-fly key-presses: • toggle with between random-wiring and regular CA. • try the 3-way toggle j for input-histogram values plotted together withthe input-entropy plot. • key-press for a new random initial state. This is also required if thedynamics stops — reaches and attractor[3]. • key-press c and e to contract/expand space-time patterns. Contractto allow more room for the entropy-density scatter plot below.(5) Toggle showing the entropy-density scatter plot with u ( • key-press , (comma) to toggle linking successive dots as in figure 5(c). • key-press ? (question mark) to toggle re-randomising at each time-step, which slows down iteration. • key-press “ (inverted coma) to toggle showing running data of thewave-length (wl) and wave-height (wh) in the terminal as in section 7.To save the data refer to s . The input-histogram (and entropy-density scatter plot) will stop updating. • key-press ; (semi-colon) to toggle showing the density return-map( • key-press , (comma) to toggle linking successive dots as in figure 6(c). • To save the density return-map data refer to • key-press to toggle space-time patterns colored by value or by neighborhood/input-histogram colors. • key-press t to toggle between a 2D movie and 2D vertical space-timepatterns as in figure 16, then to toggle upward scrolling. • when in normal 2D, key-press to toggle space-time patterns scrollingdiagonally. • key-press < to slow down iteration, > to revert to normal speed.(8) Key-press q to pause at any time for a top-right window providing optionsdescribed in • enter net- n to re-set the random wiring ( key-press q as necessary to backtrack up the DDLab prompt sequence. • key-press d to save wave-length data, or density return-map data( To run an entropy-density plot for sequential updating one cell at a time atrandom positions[1], as for the k =7 rules in figure 14, amend steps in section 10.1as follows, • After step (7) enter return until the top-right prompt revise from: ( u for a top-right updating window ( • Enter p for a top-right partial order updating window ( min: and max: , enter . • at the entropy/density: prompt ( e as in step (8).Follow further steps as listed, then toggle showing the entropy-density scatterplot with u as in step (5) in sections 10.2. You will see single-cell updates inthe 2D pattern, and a very slow trace of the plot. For other “asynchronousand noisy updating” options refer to (
11 Issues to explain the CAP model
Explaining the CAP model is work in progress. The emergence of gliders in CAcannot be predicted directly from a rule-table, so in this sense the mechanismsare unresolved — they can only be observed by experiment, but must entail feed-backs driving a glider’s head and eroding its tail. A general theory to resolvethis question would shed light on the underlying principles of self-organisation.Pulsing when the wiring is randomised must utilise the same feedbacks, but dis-tributed throughout the network instead of localised in a regular neighborhoodto create and move a glider.Future work should also address the following issues, • Because the pulsing waveform (shape/phase, wavelength, waveheight) isobserved to be diverse, how does the type of glider dynamics relate to thewaveform. • Study how the pulsing waveform breaks down as the random wiring reachis reduced — is there a phase transition? • There is a very high probability that gliders imply pulsing, but the few ob-served exceptions should be examined, gliders/no pulsing, and pulsing/nogliders. • Study how the sequential updating of one cell at a time at random positions[1]can results in pulsing. 16
In a world where much biology is produced by reproducing patters, producesreproducing patterns, or recognises these patterns, there has historically beenfocus on the chemistry and physics of thermodynamic equilibrium more so thanon the bio-physics of collective oscillatory phenomena. Oscillations can be foundin all forms of life[12], but we have focused on mammalian biology, and aspectsof human physiology where oscillations play a crucial role[1].Athough differential equation models of ocsillations in single cells have beenproposed, such as the Hodgkins Huxley equations, negative feedback with a timedelay, or coupled negative and positive feedback[13], currently there is no satis-factory theory to explain essential oscillations in whole organs, for example theheart beat, uterine contractions in childbirth, and various rhythmic behaviourscontrolled by the central pattern generators of the central nervous system, suchas breathing and locomotion[1].We are left searching for pacemaker neurones, pondering how signalling inbiofilms can occur faster than diffusion, how synchronisation can occur overconsiderable distance and how biology is so robust with such inbuilt redundancy.We propose that clusters of excitable tissue are able to oscillate according toa their appropriate waveform because non-localised connectivity[14, 15] is sub-jected to a specific rule of communication, the bio-rule, analogous to a gliderrule. The bio-rule is based on three (or more) cellular states, firing, refractory,and ready to fire, generated by chemical signalling[16], action potentials, cal-cium and sodium ion channels and concentrations. This model is favoured byevolution because a variety of synchronized waveforms can arise from differentbio-rules. Furthermore, a given waveform is independent of the exact connec-tion network, is noise tolerant, can be turned off and on by altering the reach ofnon-local connectivity, and is robust to noise, cell loss, and functional reserve.The CAP model carries the ability and benefits of modelling multiple cellssimultaneously forming a platform to probe the relationship between networkconnectivity at one level and collective behaviour at another. This summeryof the more detailed reasoning presented in [1] suggests that the CAP model isapplicable as a conceptual model for bio-oscillations and can provide a basis forfurther development of the ideas.
13 Summery
We have presented further evidence and results for this surprising phenomenonof spontaneous, sustained and robust rhythmic oscillations, pulsing dynamics,when random wiring is applied to a 2D “glider” rule running in a 3-value total-istic CA.We have reiterated the potential of the CAP model to provide a much neededdecentralised model for bio-oscillations in nature, specifically in the case of mam-malian excitable tissue.We have defined the system’s architecture, and identified the behaviour for17oth glider dynamics and pulsing, which are intimately related, and noted theissues that require explanation. A guide to the relevant functions in the softwareDDLab to repeat and extend pulsing experiments is provided. However, theunderlying mechanisms remain unresolved and are open to further study andresearch.
14 Acknowledgements
Experiments and figures were made with DDLab ( )— where the rules and methods are available, so repeatable[10]. Thanks toInman Harvey for conversations regarding asynchronous updating, to TerryBossomaier for exchanges regarding phase transitions, and to Paul Burt andMuayad Alasady for comments regarding bio-oscillations.
References [1] Wuensche,A., E.C.Coxon, “Pulsing dynamics in randomly wired glider cellularautomata”, https://arxiv.org/abs/1806.06416 , 2018, to appear in Journal ofCellular Automata.[2] Wuensche,A., A.Adamatzky, “On spiral glider-guns in hexagonal cellular au-tomata: activator-inhibitor paradigm”,
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