Group Title: Theoretical Biology and Medical Modelling 2006, 3:2
Title: Memory in astrocytes: a hypothesis
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Theoretical Biology and Medical

Modelling BMed Ctra


Memory in astrocytes: a hypothesis
Robert M Caudle* 1,2

Address: 'Department of Oral and Maxillofacial Surgery and Diagnostic Sciences, University of Florida College of Dentistry, Gainesville, Florida
32610, USA and 2Department of Neuroscience and the McKnight Brain Institute, University of Florida College of Medicine, Gainesville, Florida
32610, USA
Email: Robert M Caudle*
* Corresponding author

Published: 18 January 2006 Received: 16 December 2005
Theoretical Biology and Medical Modelling 2006, 3:2 doi:10.1 186/1742-4682-3-2 Accepted: 18 January 2006
This article is available from:
2006 Caudle; licensee BioMed Central Ltd.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Background: Recent work has indicated an increasingly complex role for astrocytes in the central
nervous system. Astrocytes are now known to exchange information with neurons at synaptic
junctions and to alter the information processing capabilities of the neurons. As an extension of
this trend a hypothesis was proposed that astrocytes function to store information. To explore this
idea the ion channels in biological membranes were compared to models known as cellular
automata. These comparisons were made to test the hypothesis that ion channels in the
membranes of astrocytes form a dynamic information storage device.
Results: Two dimensional cellular automata were found to behave similarly to ion channels in a
membrane when they function at the boundary between order and chaos. The length of time
information is stored in this class of cellular automata is exponentially related to the number of
units. Therefore the length of time biological ion channels store information was plotted versus the
estimated number of ion channels in the tissue. This analysis indicates that there is an exponential
relationship between memory and the number of ion channels. Extrapolation of this relationship
to the estimated number of ion channels in the astrocytes of a human brain indicates that memory
can be stored in this system for an entire life span. Interestingly, this information is not affixed to
any physical structure, but is stored as an organization of the activity of the ion channels. Further
analysis of two dimensional cellular automata also demonstrates that these systems have both
associative and temporal memory capabilities.
Conclusion: It is concluded that astrocytes may serve as a dynamic information sink for neurons.
The memory in the astrocytes is stored by organizing the activity of ion channels and is not
associated with a physical location such as a synapse. In order for this form of memory to be of
significant duration it is necessary that the ion channels in the astrocyte syncytium be electrically in
contact with each other. This function may be served by astrocyte gap junctions and suggests that
agents that selectively block these gap junctions should disrupt memory.

Background system. This view has now been supplanted by a more
Until recently astrocytes were considered to play no more active participation of astrocytes in information process-
than a supportive role for neurons in the central nervous ing, where the astrocytes not only receive and respond to

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Open Access

Theoretical Biology and Medical Modelling 2006, 3:2




Rule232 m O mo m mo D O D0 0 0

Figure I
Memory in cellular automata. A sixteen unit one dimensional
cellular automaton was constructed using binary units and
Wolfram's rule number 232. This rule is illustrated at the
bottom of the figure where the three squares on top are the
current states of three adjacent units and the single square
below is the resultant state of the middle unit during the next
iteration. Open squares indicate state 0 and filled squares
indicate state I. The initial representation (RI) was generated
by randomly setting the state of each unit to either 0 (open)
or I (filled). The time series was then calculated. Note that
the memory of this system extends from RI to R4 where the
representations change with each iteration. Starting at Rothe
units no longer change state indicating that all information
about R, is lost.

neuronal input, but also transmit signals to neurons [ 1-9].
These findings indicate that astrocytes contribute to the
processing of information. In support of this concept it
was recently demonstrated that spinal cord astrocytes are
necessary to support hyperalgesia produced by peripheral
injury [10-12]. Blocking gap junctions in the astrocytes
suppressed hyperalgesia, which suggested that the astro-
cytes were processing the nociceptive information and
regulating the function of spinal cord neurons [10]. These
results are similar to work reported by Hertz et al. and Ng
et al. who demonstrated that astrocytes are critical for the
establishment of learned behaviors [13,14]. Furthermore,
recent studies indicate that several general anesthetics sup-
press the function of astrocyte gap junctions at concentra-
tions that are relevant for loss of consciousness [15,16].
These data suggest that the anesthetic properties of these
agents may be mediated at least in part by their actions on
astrocytes and may indicate some role for astrocytes in

In a recent review Robertson outlined an astrocentric
hypothesis of memory [17] as an alternative to the current
neurocentric or synaptic based theories. In this hypothesis
Robertson concludes that because astrocytes form large
syncytium via gap junctions and that they are connected

to neurons through synapses these cells can store and
"bind" diverse information. In this intriguing review Rob-
ertson hypothesizes that information is stored as a result
of gap junctional plaques converting to a crystalline con-
figuration that is a closed, high resistance, state of the gap
junctions. As a result of these altered gap junctions ion
flow between astrocytes is restricted resulting in a func-
tional memory.

In examining the idea that astrocytes might play a major
role in information processing it seemed prudent to exam-
ine other potential memory mechanisms that could sup-
port information processing in astrocytes. In experiments
examining electrical potentials and calcium fluxes in
astrocytes it was demonstrated that these cells can, on an
individual basis, support potentials for several seconds
[1,2,6,7]. These data suggest that ion channel activity in a
group of gap junction linked astrocytes could retain infor-
mation for substantial periods of time. Thus, the ion
channels mediating the astrocyte potentials could func-
tion to store and process information in the central nerv-
ous system. This paper examines the possible role of ion
channels in storing information in astrocytes.

Results and discussion
Similarity of ion channels to cellular automata
Ion channels communicate with each other via changes in
voltage, changes in calcium concentrations or through
other second messenger systems. In voltage gated ion
channels, for example, the rules governing the relation-
ship between channels specify that if neighboring chan-
nels alter the local membrane potential to some threshold
the channel under observation will change state, i.e. open
or close. Each ion channel functions as an independent
unit that monitors information transmitted from its near-
est neighbors. As a result of the information processing
occurring at the single ion channel level ensembles of ion
channels are capable of performing relatively complex
functions, such as the generation of action potentials. This
form of information processing by ion channels is
remarkably similar to models known as cellular automata
[18,19]. In cellular automata simple units that are capable
of existing in a finite number of states are linked together
using rules for the transfer of information between the
units. The states occupied by the units and the rules of
information transfer determine what state each unit will
occupy in the next time period. These models have been
extensively studied and demonstrate the emergence of
complex behavior [20,21]. Some cellular automata have
even demonstrated universal computation [22]. To illus-
trate how a cellular automata stores and processes infor-
mation a one dimensional cellular automaton in which
the units are binary (they are either in state 0 or state 1) is
presented in Figure 1. The rule used was the mean of three
units rounded to the nearest integer determines the state

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Theoretical Biology and Medical Modelling 2006, 3:2

of the middle unit in the next iteration. This model was
studied at length by Wolfram and this rule is Wolfram's
rule number 232 [20,21]. In figure 1 the initiating event
(Representation 1 (RJ)) was produced by randomly set-
ting the states of the units in the automata. The time series
was then calculated. In the figure it is evident that from R,
to R4 the automaton changes representations, but after R4
the cellular automaton reaches a steady state and the rep-
resentations no longer change. This stabile representation
is the attractor Ro. The transition period from R, to R0 is
the memory of the automaton. At each iteration prior to
R0 the automaton retains information that can be used to
determine something about the initial configuration.
However, when the automaton reaches Ro all information
about the initial configuration has been lost. In astrocytes
the ion channels in the membrane are distinct units with
a finite number of states and they communicate with each
other through a simple set of rules, i.e. a change in voltage
or in Ca2+ concentration. Therefore, the astrocytes' mem-
brane ion channels are acting as a two dimensional cellu-
lar automaton. As with the automaton presented in figure
1 the initiating event can be inferred based on the config-
uration of the entire ensemble of ion channels up until
the ion channel configuration returns to the attractor rep-
resentation (Ro). At this point all information about the
initiating event is lost. This concept suggests that ion
channels working in collection can store information for
at least brief periods of time. The remaining question is
the maximum duration of memory in this type of system.

Memory in cellular automata
In a series of interesting experiments Langton examined
the properties of cellular automata that optimize informa-
tion storage and processing [23]. In these experiments he
varied the rules by which the cellular automata operated
and measured the resulting chaotic nature of the system.
Langton found that automata whose rules made them
operate at the junction between ordered and chaotic
behavior were able to store information for the longest
period of time. Memory dropped off markedly on either
side of this phase transition. To illustrate how the chaotic
nature of the cellular automata might influence memory
a two dimensional cellular automaton with four different
rule sets and a Moore neighborhood (8 neighbors) was set
up (Figure 2A). The units in the automaton could occupy
four different states, i.e. one open, one closed and two
inactive. The cellular automaton was seeded with two
units in the open state to invoke the initial representation
R1. The left hand column illustrates a rule set that pro-
duces ordered behavior. Note that a signal cannot propa-
gate in this cellular automaton. The second column
demonstrates another form of ordered behavior where the
behavior immediately becomes repetitive. This cellular
automaton, like the one to the left of it, cannot process
information due to the inability of the automaton to tran-

sition to novel representations. The third column is a rule
set that produces behavior at the border between order
and chaos. The net result is the smooth propagation of an
"action potential" throughout the cellular automaton
with the system eventually returning to the attractor rep-
resentation Ro. The final column illustrates a chaotic sys-
tem that evolves rapidly into a random pattern of channel
openings. The nearly random behavior prevents proper
processing of information since there is no relationship
between successive representations. Figure 2B illustrates
the "potentials" produced by these different rule sets by
plotting the number of open channels versus time. These
models demonstrate that only the rule set with behavior
at the transition between order and chaos produces a
potential that is similar to an action potential observed in
biological systems. Note that the rules that produce
ordered behavior either returned to the attractor represen-
tation Ro very rapidly or never returned to R0, suggesting
that the systems are incapable of supporting information
storage. The chaotic rule set also never returns to the
attractor, which also indicates that the system cannot
retain information for significant periods of time. Only
the rule set that produced behavior between order and
chaos could retain information about the initial event R,
for a period of time and then return to the attractor repre-
sentation. Based on the similarity of the potentials gener-
ated by the transition rule set these models suggest that
the ion channels in the membranes of biological cells
function as cellular automata with rules that set the
behavior at the boundary between order and chaos. This
region of the order to chaos spectrum balances informa-
tion storage with transmission, which, in turn, supports
information modification [23].

In addition to examining the length of memory in cellular
automata relative to the chaotic nature of the automata,
Langton [23] evaluated how the number of units in an
automaton influenced memory. In these experiments
Langton used rules that produced automata that operated
in the order/chaos phase transition and then varied the
number of units in the automata. He found that there was
a log-linear relationship between the time that the cellular
automata stored information and the number of units in
the automata. This indicated that the addition of units to
the automata exponentially increased the amount of time
the automata stored information. This relationship is an
extremely powerful property of cellular automata that has
evolutionary significance for biological systems that proc-
ess information with ion channels. The exponential rela-
tionship between memory and the number of units in an
automaton indicates that a biological system simply has
to add more units (ion channels) to its calculating device
in order to dramatically increase its memory. With an
increase in memory duration the complexity of the calcu-
lations that can be performed also increases [23].

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Theoretical Biology and Medical Modelling 2006, 3:2

Ordered Ordered Transition Chaotic





Ti 0

10 0 10 20 30 40 50 60 70 80 90 100


10 0 10 20 30 40 50 60 70 80 90100


10 0 10 20 30 40 50 60 70 80 90 100

30 0 102030405060 80 90


10 0 10 20 30 40 50 60 70 80 90 100

Figure 2
Two dimensional cellular automata operating between order and chaos behave like excitable membranes in biological cells. A.
A two dimensional cellular automaton was constructed with the program CaSim using units with four states, i.e. one open, one
closed and two inactive states. Four different rule sets were used to generate the four time series in the figure. The cellular
automaton was seeded at R, by setting two units to the open state and the times series calculated. The configuration of the cel-
lular automata at iterations 0, 1, 25, 50 and 100 are presented in the figure for the four rule sets. The entropy of the rule sets
was determined by calculating the probability of each state (Ps) from 10 runs of 1000 iterations. For these calculations 10 per-
cent of the units were set to the open state at R1. Entropy was calculated using the equation: entropy = -Z Ps In (Ps). The
entropy of each rule set was then expressed as a ratio of the calculated entropy to the maximum entropy (bottom of the fig-
ure). The maximum entropy is when all four states have a probability of 0.25. B. The "potentials" generated by the rule sets in
A were graphed by plotting the number of open channels versus time. These plots indicate that only the transition rule set pro-
duces channel openings that are similar to action potentials in biological membranes.

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Theoretical Biology and Medical Modelling 2006, 3:2

-37 T-------1---
0 10000000 20000000
# Ion Channels

Figure 3
Memory as a function of the number of
was collected from the literature for the
for single ion channels, the length of pot
gle cells and the length of potentials in gr
slices or mollusk ganglia. The logs of the
and single cell data were graphed versus
channels. Cells were estimated to have I
slope of the line defined by these two po
and the length of the potentials in the br
lusk ganglia were plotted onto the graph

The human cellular automaton
The findings of Langton indicate that
aton is increased in size the duration o
In the astrocentric hypothesis large nu
are connected through gap juncti
which suggests that astrocytes form ex
cellular automata. To examine the
duration for a human brain sized cellu
was collected from the literature for n
nel open and closed times, duration c
in single cells by very brief stimuli a
potentials in brain slices and mollusk
ings in the slices and ganglia used for
sented a large number of cells in the
single cell in the slice or a population
synaptic event. Since data are limi
potentials from all forms of excitable
In figure 3 the log maximum length
single ion channels to transition thr
closed cycle and the log of the duratic
cell potentials were plotted versus t
channels. For whole cells the number
estimated to be 106. A regression line
two sets of data. The duration of poter
and ganglia were then plotted on this 1
of ion channels needed to produce t
estimated by extrapolation. These pot
be generated by 107 to 108 ion channel
gests that Langton's relationship of the
length of time that information is

automata holds true for ion channel cellular automata.
Note that for convenience there was no attempt to limit
the data collected to any one type of ion channel, cell type,
or species. The assumption used here is that all biological
systems evolved a similar mechanism to process informa-
tion with ion channels and, as such, their ion channels
Ion Channels have similar properties.
Slices and Ganglia
To generate an estimate of the total number of ion chan-
nels in a human astrocyte cellular automata the number
30000000 40000000 of astrocytes was approximated to be 1013 [28]. With 106
ion channels/cell this suggests 1019 ion channels in a
human cellular automaton. Using the estimate of 1019 ion
channels in the human cellular automaton the predicted
ion channels. Data duration of memory was extrapolated from the slope of
openlosed times the line in figure 3. The relationship between memory and
entials evoked in sin-
entils evoke in s-in- "- f ion channels was estimated to be
oups of cells in brain (2.3X10-N)
single ion channel t = e(21 N). Where t is time and N is the number of
the number of ion ion channels in the system. This calculation yielded a pre-
06 ion channels. The dicted maximum men --- -- human sized astrocyte cel-
ints was determined lular automaton of 10(102) years. Therefore, for all
ain slices and mol- practical purposes, the predicted maximum duration of
memory in human cellular automata is infinite. What is
most notable about this memory is that it occurs without
fixing the information to any physical structure such as a
synapse or cell as predicted in Hebb's postulate [29]. The
as a cellular autom- information is stored as a succession of representations, or
f memory increases, ion channel configurations, with each individual repre-
imbers of astrocytes sentation lasting only a short period of time. The configu-
ons [10,17,24-27], ration of the ion channels is organized by the incoming
tensive ion channel information and then as this organization dissipates over
potential memory time the information is lost. In thermodynamic terms the
lar automaton data entropy of the system is decreased by the storage of infor-
laximum ion chan- mation and, as the calculation presented above indicates,
)f potentials evoked it takes a substantial amount of time for the entropy to
nd the duration of return to baseline levels. Admittedly, the estimates for the
ganglia. The record- number of ion channels and the number of astrocytes that
this analysis repre- make up a single syncytium are crude; however, even if the
tissue rather than a estimates are off by several orders of magnitude the over-
response to a single all conclusion that the potential duration of memory in a
ted for astrocytes, human ion channel cellular automaton is infinite, from a
cells were collected, biological frame of reference, remains valid.

of time reported for
ough an open and
on of evoked whole
he number of ion
of ion channels was
was fitted to these
itials from the slices
ine and the number
these potentials was
:entials appeared to
Is. This finding sug-
number of units to
stored in cellular

Another interesting comparison to be made between the
astrocentric hypothesis and the neurocentric hypothesis is

that there are k(10') distinct representations or unique
configurations of the ion channels. Using 1012 neurons
each possessing 103 synapses we can estimate that there
are 1015 synapses in a human brain [28] and a potential

for k(15) distinct representations or unique configura-
tions of the synapses. The term k is the number of states
that an individual ion channel or synapse can take. These
calculations demonstrate that the potential information

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Theoretical Biology and Medical Modelling 2006, 3:2

processing capacity of the astrocytes using ion channels is
many orders of magnitude larger than the capacity of neu-
rons using synapses.

Associative memory in cellular automata
An important component of memory is the ability to asso-
ciate two or more events. In an ion channel cellular
automata this is accomplished by the fact that the series of
representations produced by a single event is significantly
different from that produced by two events. Figure 4 dem-
onstrates the ability of a cellular automaton to associate
information from two events. In the first column a single
event produces a series of representations as the automa-
ton progresses. In the second column two events occur
simultaneously. The two events produce a series of repre-
sentations that are distinct from the single event presented
in the left column. This indicates that the two events have
been associated to produce a unique memory.

Another interesting facet of ion channel cellular automata
is that because they are dynamic systems they can readily
store temporal differences between events. In the right
hand column the two events are separated by ten units of
time resulting in a series of representations that differs
from either the single event in the left hand column or the
two simultaneous events in the middle column. These
observations suggest that the proposed astrocyte memory
system can associate memories and that temporal infor-
mation can be stored.

Research supporting astrocyte cellular automata as
memory systems
In studies published over forty years ago Hyden demon-
strated that glia were critical for memory [30-32]. More
recent work using the one-trial aversive learning paradigm
in chicks has confirmed Hyden's findings [13,14,33]. In
these studies inhibitors of astrocyte function were found
to block both short term and intermediate term memory,
but, when administered later, had no effect on the long
term retention of the learned behavior. During the short
and intermediate periods it was demonstrated that ion
fluxes in astrocytes are critical [13,33,34] for memory sug-
gesting that the astrocyte ion channels may store informa-
tion in the chicks for a brief period of time, approximately
60 minutes, while the appropriate rewiring of the neuro-
nal circuitry takes place. It is important to note that this
behavioral model involves both memory and learning,
while the cellular automata hypothesis presented here is
related purely to memory. Memory is the ability of an
organism to store information about events in a retrieva-
ble format, whereas learning involves a change in behav-
ior or potential behavior. Thus, a consolidated learned
behavior, as occurs in the one-trial aversive learning para-
digm, is likely to be the result of neuronal rewiring. Fur-
thermore, it does not require the organism to retain any

specific memory of the event that precipitated the change
in behavior beyond the length of time necessary to pro-
duce the rewiring. In this light, the chick in the aversive
learning paradigm may actually recall the aversive stimu-
lus for the short and intermediate term memory periods,
which require astrocytes, but may not retain any recollec-
tion of the event once the aversive behavior has been
established. It is enough for the chick to avoid certain
objects without remembering why it needs to avoid them.
The distinction between memory and learning is impor-
tant because the two processes are likely mediated by dif-
ferent mechanisms. In the current hypothesis the ion
channel cellular automata would be responsible for the
specific memory of the event while changes in synaptic
strength of the neurons would be responsible for learning
and maintaining the new behavior. Astrocyte memory
could support learning, but learning does not necessarily
support the memory of events.

In addition to proposing that glia were involved in mem-
ory, Hyden predicted that mental diseases may involve
glia [35] as reported in [34]). In the ion channel cellular
automata hypothesis it is critical that the ion channels
operate at the junction between order and chaos. Depar-
ture from this behavior is predicted to produce pathology.
Deviation to the ordered side of the spectrum might pro-
duce depressive types of behaviors in the organism and
memory deficits while deviation to the chaotic side might
produce psychotic or manic types of behaviors that are
also associated with memory deficits. Several studies have
demonstrated that long term treatment with antidepres-
sant drugs at clinically relevant doses alters protein expres-
sion and function in astrocytes [36-41] and long term
treatment with lithium ion results in suppression of
mRNA for sodium-dependent inositol transporter in
astrocytes [34]. The length of treatment required for the
change in astrocyte proteins is consistent with the onset of
the therapeutic effect of these agents. These studies suggest
that these psychoactive agents may adjust the activity of
astrocyte ion channel cellular automata toward the order/
chaos border, thus improving the function of the memory
system. Therefore, a number of studies, spanning over
forty years, indicate that astrocytes are important for
memory and possibly for the therapeutic effect of psycho-
active drugs, which is consistent with the astrocyte ion
channel cellular automata hypothesis.

In this study the hypothesis that astrocytes could store
information in the central nervous system was considered.
Based on the similarity of membrane ion channels to
mathematical models known as cellular automata it
seems reasonable to conclude that ion channels in astro-
cytes could store information for significant periods of
time. This storage system does not rely on physically fix-

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Theoretical Biology and Medical Modelling 2006, 3:2

Single Event

Two Events

Two Events
Temporally Separated

Figure 4
Associative memory in cellular automata. A cellular automaton operating at the transition between order and chaos was setup
as described in figure 2 using the program CaSim. Three different stimuli were used. Iterations 1, 14 and 25 are presented in
the figure. In the left column the cellular automaton was seeded by setting one unit to the open state at R, (Single Event). In the
center column two units were seeded at R, (Two Events). In the right hand column the cellular automaton was seeded by set-
ting one unit to the open state at R, and a separate unit to the open state at Rio (Two Events Temporally Separated). Note that
each time series generates a different pattern of channel openings (representations) indicating that the two events in the sec-
ond and third columns have produced unique memories by associating the events. Also note that the difference in representa-
tions produced by the automaton in the second and third columns indicates that the cellular automaton stores temporal
information about the events. Therefore it is concluded that a two dimensional ion channel cellular automata is capable of asso-
ciative memory.

ing information to any structure such as a synapse; rather
information is stored by organizing the activity of the ion
channels. If this concept is correct it suggests that neurons
may use astrocytes as a dynamic information sink. In the-
ory, this information would remain readily available to

the neurons for extended periods of time. Furthermore,
this hypothesis indicates that to store information for sig-
nificant periods of time the ion channels in the astrocyte
syncytium must be in electrical contact with each other.
This function could be served by the astrocytes' gap junc-

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Theoretical Biology and Medical Modelling 2006, 3:2

tions. Thus, we can predict that agents that selectively
block astrocyte gap junctions should disrupt memory.
Clearly, further work is needed to verify this theoretical
framework for memory in nervous systems.

One dimensional cellular automaton
A 16 unit one dimensional cellular automaton was set up
with each unit having 2 states. The rule used for this
automaton was Wolfram's rule number 232 [20,21]. In
this rule each unit is updated by averaging the states of the
unit with its two nearest neighbors and then rounding to
the nearest integer. The time series for this cellular autom-
aton was calculated by hand.

Two dimensional cellular automata
To examine the effects of different rule sets on 2 dimen-
sional cellular automata the program CaSim [42] was
used. A matrix of 100 x 100 units with a Moore neighbor-
hood (eight neighbors) was set up with various rules. Each
unit had 4 states. The entropy of the different rule sets was
calculated using the equation entropy = -1 Ps In (Ps),
where P, is the probability of a unit occupying a particular
state. The probabilities of the different states were deter-
mined from 10 runs of 1000 iterations for each cellular
automaton. For these calculations the cellular automaton
was seeded for each run by randomly setting ten percent
of the units to the open state. The maximum entropy was
calculated using the probability of 0.25 for each of the
four states. The ratio of the calculated entropy of the rule
set to the maximum possible entropy was used as an indi-
cator of the chaotic nature of the system. Thus an entropy
ratio of 0 is a completely ordered rule set and a ratio of 1
is a completely chaotic rule set.

For the examples presented in the figures the cellular
automata where seeded with either 1 or 2 units set to the
open state.

Duration of memory versus the number of ion channels
To calculate the relationship between the number of ion
channels in a system and the duration of information
storage by the ion channels data was collected from pub-
lished sources. The maximum open and closed times for
various ion channels were obtained [43-54] and the open
to closed cycle was used as the duration of memory in sin-
gle ion channels. Similarly, potentials recorded in single
cells were obtained [55-62] and used as an indication of
the activity of multiple ion channels in concert. The log of
the values for the duration of the responses in the ion
channels and cells were plotted versus the number of ion
channels. The number of ion channels in the cells was
estimated to be 106. A line was then fitted to the two
points and the log of the duration of potentials in slices
and ganglia [63-71] were plotted on the line.

Competing interests
The authors) declare that they have no competing inter-

This work was supported by the University of Florida College of Dentistry
and the McKnight Brain Institute.

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