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Model of the Synaptogenesis: Stochastic Graph Grammars
MODEL OF THE SYNAPTOGENESIS: STOCHASTIC GRAPH GRAMMARS Maurice Milgram UniveTsile de Compiegne, BP 233, 60206 Compiegne Cedex, France
ABSTRACT
tion of the device. Here we would like to say a word about the stochastic aspect of the evolving network models. It is often true that one uses a probabilistic approach to make up for a lack of data in the systems studied. It has already been shown that the presence of noise or of uncertainty at a given level can involve an increase in order at the upper level. We are reminded of Von Foerster [9] and H. Atlan's [10] work who showed that a system provided with certain constraints could evolve, under the influence of an outside noise, towards states more structured, more organized. We try to develop this idea of 'organization from noi se' in the case of synaptogenesis. The synaptogenesis, that is a creation of connections between neurons, is a biological process which is not yet completely understood . We cannot therefore expose physiolo.Iigical theories that are presently proposed and the experiments which they are based on . One can rough ly reduce the knO\.,l edge about cs: synaptogenes i s to these fo 11 owi ng s tatemen:nce [ 5,5 ] ~ te des) 1) Connections between neurons (synaps' and exist under three states: labil, stable ~ inpu·~he degenerated . The connection has to be in 'creasEd 1abil state before the stable one ; using )n has information from neurons that are concerne ace' with this connection, it can then become 'to i ' stable or degenerate. 's 2) The maximum wiring of the neural net~w Cf work and the main stages in its evolution :e . are geneti cally programmed: the detail of' +'J~e these connections being based on both chas~ ~ and the functioning of the network (the l~ ce ter being 90verned by the internal and ex">,f a~_ ternal enVl ronment). fn3) The learning capacity of a neuron ne1 work is limited to the variability among the group of connections (synap$es) inside a genetically programmed 'envelope'. It appears that chance is used here to build the delicate structure of the neuron network under control of biochemical mecanisms which select the useful connections. Nature, thus, economizes in the enormous programming entailed with each connection. This plasticity (at a certain level) is also used by the network as a means of adaptation (learning by experience). This requires a certain redundance, at least to start with; such an initial redundance was actually ob-
In this paper, we present a review of the principal models of synaptogenesis. We try to describe a new model which we call 'evolving cellular system'. This model is composed of two elements : a webs grammar and a labeled network. The network labels are determined at each node, by a stochastic automaton . The network is self-constructing by means of the webs-grammar and the local stochastic automata. I NTRODUCTI ON We propose a review of the principal mathematical models which describe evolving networks, that is to say a group of interconnected cells whose number is variable in time . Let us now talk about the motivations of such models. All living organisms are composed of cells which are born and after specialize progressively. Certain animal societies (such as an insect society) can therefore be considered as super-organisms whose cells are the members of the society. The notion of growth in a cellular system implies the hypothesis that all cells (at least to begin with) are identical. The problem is to describe the ways in which the cellular system acquires a form and can fulfill a given function. We shall name these means: growth program. The complexity of the final result (the constituted cellular network) is sometimes very big if one compares it to the complexity of . the growth program. In the case of a li vi ng organism, the base of such growth program is contained in the genetic memory of a cell. We are concerned with trying to construct a model by use of reduced growth program which permits us to obtain complex networks using hypotheses which can be realized biophysically. In spite of extreme simplification of reality, certain models bring new light to certain questions. If we take the example of Winograd and Cowan's work [8] on 'computation in presence of noise', it is clear that the simplification of neurons is enormous, nevertheless, th~ authors prove that it is possible to reallze under certain hypotheses a reliable computation device from weakly reliable modules. This result remains true even if we admit the possibility of errors in the construcI.s.- M
I7I
172
M. Nilgram
served.
cell to others. See Fig. 1 for more details. I. MODELS OF THE SYNAPTOGENESIS
Several authors have proposed mathematical models of the synaptogenesis. Some of them have been made precisely for this aim while others, more general, can be used too. Courrege, Changeux and Danchin [5,6] have proposed a very abstract theory of the epiyenesis of neuronal networks by selective stabilization of synapses. They introduce a formalism to represent an evolving neuronal netvlOrk and the effects of envi ronment. These effects are obtained from stabilization or degeneration of labil synapses associated with functioning. Learning is related to a variability of the connective organization of the network : the interaction of the environment with the genetic program is memorized as a particular wiring through neuronal functioning. Authors apply this theory to the development of the neuromuscular junction. This theory may lead to a lot of specific models and requires for actual utilization precisions about environment and all the parameters that are involved in the functioning of the network. It rather provides a framework for particular studies and models that fulfil mathematical and biological assumptions on which this theory is grounded. We shall say just a few words about the use of cellular automata as models of synaptogenesis . Initialy created by Von Neuman for simulating, living organims, many studies have been carried off about cellular automata both on a theoretical and practical level (especially in pa~llel computing). Arthur Burks has proposed r~ ,11] a model of growing automata inspired [3,the two-dimensional cellular automata . of , " model is dedicated to the computer Th~Slce but it contains a very interesting ~~le~Ssion about the measure of the comp1e~sc of an automaton. In fact, cell ular auxlt;ata seems to be inadequate to describe torr synaptogenesis (even schematically) thE;ause of the spatial uniformity of its beSnective organization. We should deal with ~Ol at another 1eve 1, a phys i ca 1 cell bei ng 1 t ]resented by a great number of the cell s rerthe automaton. of Jt ten years ago, M.J. Apter [2] deAbol Jped a model of a 'self-constructing ve1r, which will be used as a starting-point net,ow. The basic idea of the model is the ~el110Wing one: at the beginning there is ,?ust one cell, this cell can reproduce it~self, the new cell will be identical to the bld one. The programmation of the cell is made up of a directed graph (printed for commodity inside the cell). This graph contains several categories of edges and vertices. Pulses (signals) move along edges (channels) and when a pulse enters a special node (reproduction node labe1ed by the letter R) the whole cell reproduces itself (a new cell is produced with an identical network of each cell is connected to others. Hence there is no di fference between internOl pulses and external pulses from one '
-~1-
pulse
I
®----rl) ~---E9The system is fully deterministic. The final pattern only depends upon the net of the initial cell. As there is no external information (input), we can say that we have a self-constructing network. The position of a dauthter-ce11 is entirely determined by the position of its mother-cell: in the same way the connection of the daughter cell are necessary carried off with the mother cell. In Apter's model position and connection of a cell is subordinated to its mother cell, hence depends on the reproduction process; this constitutes a practical handicap and is not justified from a neurophysiological stand point. Apter then has to improve this model, especially by assuming the existence of an internal clock capable of synchronizing some operations. Some nodes of the net are activated at precise moments; in keeping with the existence of
Model of the synaptogenesis
maturity period which has been experimentaly proved in the development of the central nervous system. It increases highly the expanding possibilities of the network. The author introduces the notions of 'input' and 'output' nodes allowing a change in the wiring of the network different from that consisting in reproducing a cell. One car: justify this method (or at least its principle) by neurophysiological experiments in which chemioaffinity seems to play a prominent part in synaptogenesis. A certain neuron (say A) sends out a specific chemical substance ; if another neuron (say B) is in a certain state and receives this signal, a new connection between A and B will be created. In fact, Apter's model is designed to prove that it is possible to make up highly complex systems from a set of relatively simple instructions. The fact that all the cells have exactly the same structure is a tool for simplifying the construction of the network. Any cell of the net contains the set of instructions necessary to produce the whole development of the system. We are now going to present our model of an evolving network.
stochastic graph grammars
Webs grammar). ~, stands for the set of messenger states (states present in the transition function of the automata). d) SC stands for the oriented and labelled graph nwhose vertices are the cells (at step n) labelled with their states and edges created by the ~rammar. e) If A is a node of SC , ~ + (A) stands for the application ~ + N dRfinRd by ~ +(A) (m) = the number of nodes B of SC , sUch as (B,A) is an edge of SC and suchnas the state of B is n. n In the same way 14 -(A) with the mode B', such as (A,B') isnan edge of SC . The couple [M +(A) , f'" -(/I)] thUs expresses both the presgnce and ~he number of the messages carried (at phasexn) by the neighbouring cells of cell A. f) The transition function is defined by: M M 4> : [. x N x N x 'j x n + f 4>
(A, [/4n +(A), Mn- (A) ], i, w) £
t
M
1I F
In the case ~ = N, i will take for value the age of A, i.e. the number of steps gone by since A was produced (either!tn the initial CS or by Webs grammar). The ECS is defined by :
11. PROPOSED MODEL ECS = We propose a new model called evolving cellular system (ECS) which is composed of an initial cellular system (CS) and of a composed webs grammar. (See Annex). Each cell of the initial CS is equiped with a stochastic automaton that determines at each step the label of this cell (we take as new label the name of the actual state of the automaton). Webs grammar acts between two iterations of the stochastic automaton and modifies the structure of the CS (for instance: connections, number of cells, etc ... ). These two aspects of the model play very di fferent parts. The CS represents a certain physical reality and stochastic automata have to mirror this reality. The grammar is just a tool to describe the set of e vents one can obse rve in the CS. The derivation rules of such a grammar have nevertheless to take into account natural processes. By that very fact, the application of a rule depends only on the state of each cell and on the connective organization. With rega rd to each cell, the t rans iti ons from one state to another may depnd on the state of the neighbouring cells, as we shall see below. Therefore we have to deal with two different and complementary processes for modi fying the structure of an ECS. We are going now to present several definitions : a) (Il,p) is a probabilised space that models the stochastic aspect of the automaton b) ~ is an input set; we shall particul arly deal with the case ~ = N (we call it clock-input). c) E = F U M is the set of the states of a cell. F stands for the set of functional states (states present as label in
173
[t,
';1, (Il ,p),
4> ,
CS o '
G)
in which CS is the initial graph (with the initial sta£e.as label) and G a Webs grammar whose lavels are taken from the set F (FS
E)·
I f ~ = Nand 4> defi neci as shown above, the system is said 'with internal clocks'. If ~1 = 0 it is said to be 'without communication' (by convention, if M = 0, 4> : E x I x n + E) . A state eo £ F is chosen as a reference state and we set that any newly created cell is in the state e . A special convent~on concerning the input of a cell created by the grammar increases the power of the ECS. This convention has it that at the moment it is created,a cell does not have the age 0 (conversely to initial cells) but it has the age of its mother cell. The mother cell of a new cell is only defined in the following case: the grammar includes a derivation rule a + 8 with more nodes in 8 than in a, and a must be reduced to single node, and this cell (say A) is called the mother cell of all the cells belonging to 8 different from the image of A. For instance: the following rule a
CJ In the first case we can say that the cell with label x is the mother of the cell carrying the label y.
174
M. Milgram
E = { (X,d) I (X,d) an edge in the host web} U = { (d,X) I (d,X) an edge in the host web} It is a rule of cell division; the cell carrying the label 'd' is the mother of the created cell .
a
DJ
..
3)
. D
In the second case there is no mother cell of the cell carrying the label z. We are going to give a very simple example of ECS with internal clock and without communication I = N E = F = {e ,d,m,e,s,r} M = 0 The follow~ng transition graph pictures the transition probabilities between states that are different from o. The probabilities are expressed below in regard with the age n of the cell. ~~~f!1el~
:
un = Prob
{ ~ (eo,n,w)
d
d}
s
..r=:l ~
e
a
E = all the edges of the host web are kept. It is a connection rule (s = source, e = entry) . We have achieved a simulation (i.e . a realization) of this cellular system with an initial CS made up of a single cell (of age 0) that cell being in a state of rest (i .e. state eo).
m
Wn
Nom
: 0
2 3
4
5
6 7 8
:e o d d d B e0 e s C eo e e D
d s r e
d s r s e
d
9
10
r r r r r r
r r r r r r
-----:----------------------------------A
n=O n>l
un
\6
wn
1 - v n
Yn
1/ 2
rn =
!i/2
\In
_!
vn -
10
n~ 5
n;.6
x n
1 - un
tn
1/ 2
\In
zn = 1 - r n The grammar contains 3 productions 1)
a
..
D
/3=cf>
(s is the graph without nodes). This Droduction mirrors the vanishing of a cell. It supposes that the empty Web exists (neither nodes nor edges) . The embedding of the empty web (0) in the host web is immediate (all the edges such as (x,m) and (m, x) are deleted) . 2)
eo
F G
r r r r r r
Each line includes the name of a cell followed by the sequence of the states taken by this cell. The final graph (all the cells are in the state r) is the following one :
n;; 5 n?6
D
0
E
r r s r
m r r r r
~.
G
Let us notice that a deterministic ECS (i.e. with a trivial probabi1ized space (n,p) reduced to a single element) that would give a similar result with the same type of webs grammar would require a great number of states (several dozens) while ours possesses only 5 of them. nn the other hand ifwe try to lower the number of states by allowing a communication between cells, the problem put by the synthesis of such an automaton is very comple x but can be approached with original methods. I I 1. DI SCUSS ION We aim at studying evolving systems of interconnected cells. The evolution rules of our system are included in a stochastic auto~aton ( a sort of genetic program) which contains a special kind of input: either they are in relation with the
Model of the synaptogenesis
age of the cell or they are messages received from neighbouring cells. Until now we have made a few prototypes of such systems work and the final note of paragraph II reports some of these remarks. We try to check general ,.hypothesis about the fonctioning and the evolution of such systems. For instance: is it possible to generate (almost surely) an element belonging to a class of graphs choosen in advance by the means of a ECS with a bounded size? Such questions have already been put by several authors about other processes generating graphs (generally in a deterministic way) [4]. The relations between the complexity of a graph and that of the automaton that generates it must certainly be studied further. Two approaches seem to be possible : a) by the use of simulating b) by achieving a consistent theory of the ECS Experience only will allow us to choose between the 2 terms of this alternative. Acknowledgments: I am pleased to acknowledge the considerable assistance of t1rs Cindy Moreau and Mr Jean-Jacques Levive for the translation of this paper. APPENDIX: Webs grammar [1,7] A weh is a directed finite graph with labeled vertices. A web grammar G is a four ~· tuple G = (V N, VT, P, S) where VN is a set of nonterminals, VT of terminals, S a set 'initial' webs, a set of web productions or rewriting A web production (or rule) is defined Cl
....
a set and P rules. as :
S (E)
where Cl and S are webs and E is an embedding of S. More precisely E consists of a set of logical functions which specify wether or not each vertex of the host web (which does not belong to S) is connected to each vertex of s. ~~~~e!~ : G = (VN' VT, P, S) V N
= { A}
V T
and P : {(p,a)
I
{(p,A) (3) :
)
,~
)
~}
•
A (p,A) an edge in the host web}
~
(2) ~
E
= {:
S
{a}
•a
(1) •a
E
=
•A I
(p,A) an edge in the hos t web}
•
a
E is the same as in (1). The langage \L(G) generated by a web grammar G is the set of all webs that we can derive from the initial set S by successively applying productions.
17.
stochastic graph grammars
In our example A
~--~·A--~-
-- ~A_ ~o_ A
o~~o~ One can easily check that the langage generated by this web grammar consists of all directed trees which have least elements. REFERENCES [1] K.S. FU - Syntactic methods in pattern recoqnition Academic Press, NY, 1974 [2] ~1.J . APTER - The genesis of neural patterns Advances in cyhernetics and systems research, GB, 1972, J., 252-268 [3]A.W . BURKS Ed. - Essays on cellular automata Univ. of Illinois Press, Urbana, Illinois,
1970 [4] A.
LINnEN~AYER, G. ROZENBERG Ed. - Automata, languages, development North Holland Publishing Company, 1976 [5] JP CHANGEUX - A. DANCHIN - Selective stabilization of developing synapses as a mechanism for the s~ecification of neuronal networks . Nature, vol 264, December 23/30, 1976 [6] J.P. CHANGEUX - P. COURREGE - A. DANCHIN A theory of the epignesis of neuronal networks by selective stabilization of synapses Proc. Nat. Acad. Sci. USA, vol 70, N°lO, 2974-2978, Oct 197~. [7] N. ABE - M. MOZIMBTO - J. TOYODA - K. TAN AKA - Web grammars and several graphs Journal of Computer and System sciences, 7,
37-65, 1973.
[8] S. WINOGRAD - J . D. COWAN - Computation in presence of noise t1IT Press '10nograph, 1963 [9] H. VON FOERSTER - On self-organizing systems and their environments In self-organizing systems, Yovitz, Cameron Ed., Pergamon Press, ~1-50, 1960 [10] H. ATLAN - L'organisation biologique et la theorie de 1 'information Hennann, Paris, 1972 [11] A. BURKS - Computation, behavior and structure in fixed and growing automata In Self-Organizing Systems, Yovitz and Cameron Ed., Pergamon Press, 282-:~09, 1960.
ABSTRACT
tion of the device. Here we would like to say a word about the stochastic aspect of the evolving network models. It is often true that one uses a probabilistic approach to make up for a lack of data in the systems studied. It has already been shown that the presence of noise or of uncertainty at a given level can involve an increase in order at the upper level. We are reminded of Von Foerster [9] and H. Atlan's [10] work who showed that a system provided with certain constraints could evolve, under the influence of an outside noise, towards states more structured, more organized. We try to develop this idea of 'organization from noi se' in the case of synaptogenesis. The synaptogenesis, that is a creation of connections between neurons, is a biological process which is not yet completely understood . We cannot therefore expose physiolo.Iigical theories that are presently proposed and the experiments which they are based on . One can rough ly reduce the knO\.,l edge about cs: synaptogenes i s to these fo 11 owi ng s tatemen:nce [ 5,5 ] ~ te des) 1) Connections between neurons (synaps' and exist under three states: labil, stable ~ inpu·~he degenerated . The connection has to be in 'creasEd 1abil state before the stable one ; using )n has information from neurons that are concerne ace' with this connection, it can then become 'to i ' stable or degenerate. 's 2) The maximum wiring of the neural net~w Cf work and the main stages in its evolution :e . are geneti cally programmed: the detail of' +'J~e these connections being based on both chas~ ~ and the functioning of the network (the l~ ce ter being 90verned by the internal and ex">,f a~_ ternal enVl ronment). fn3) The learning capacity of a neuron ne1 work is limited to the variability among the group of connections (synap$es) inside a genetically programmed 'envelope'. It appears that chance is used here to build the delicate structure of the neuron network under control of biochemical mecanisms which select the useful connections. Nature, thus, economizes in the enormous programming entailed with each connection. This plasticity (at a certain level) is also used by the network as a means of adaptation (learning by experience). This requires a certain redundance, at least to start with; such an initial redundance was actually ob-
In this paper, we present a review of the principal models of synaptogenesis. We try to describe a new model which we call 'evolving cellular system'. This model is composed of two elements : a webs grammar and a labeled network. The network labels are determined at each node, by a stochastic automaton . The network is self-constructing by means of the webs-grammar and the local stochastic automata. I NTRODUCTI ON We propose a review of the principal mathematical models which describe evolving networks, that is to say a group of interconnected cells whose number is variable in time . Let us now talk about the motivations of such models. All living organisms are composed of cells which are born and after specialize progressively. Certain animal societies (such as an insect society) can therefore be considered as super-organisms whose cells are the members of the society. The notion of growth in a cellular system implies the hypothesis that all cells (at least to begin with) are identical. The problem is to describe the ways in which the cellular system acquires a form and can fulfill a given function. We shall name these means: growth program. The complexity of the final result (the constituted cellular network) is sometimes very big if one compares it to the complexity of . the growth program. In the case of a li vi ng organism, the base of such growth program is contained in the genetic memory of a cell. We are concerned with trying to construct a model by use of reduced growth program which permits us to obtain complex networks using hypotheses which can be realized biophysically. In spite of extreme simplification of reality, certain models bring new light to certain questions. If we take the example of Winograd and Cowan's work [8] on 'computation in presence of noise', it is clear that the simplification of neurons is enormous, nevertheless, th~ authors prove that it is possible to reallze under certain hypotheses a reliable computation device from weakly reliable modules. This result remains true even if we admit the possibility of errors in the construcI.s.- M
I7I
172
M. Nilgram
served.
cell to others. See Fig. 1 for more details. I. MODELS OF THE SYNAPTOGENESIS
Several authors have proposed mathematical models of the synaptogenesis. Some of them have been made precisely for this aim while others, more general, can be used too. Courrege, Changeux and Danchin [5,6] have proposed a very abstract theory of the epiyenesis of neuronal networks by selective stabilization of synapses. They introduce a formalism to represent an evolving neuronal netvlOrk and the effects of envi ronment. These effects are obtained from stabilization or degeneration of labil synapses associated with functioning. Learning is related to a variability of the connective organization of the network : the interaction of the environment with the genetic program is memorized as a particular wiring through neuronal functioning. Authors apply this theory to the development of the neuromuscular junction. This theory may lead to a lot of specific models and requires for actual utilization precisions about environment and all the parameters that are involved in the functioning of the network. It rather provides a framework for particular studies and models that fulfil mathematical and biological assumptions on which this theory is grounded. We shall say just a few words about the use of cellular automata as models of synaptogenesis . Initialy created by Von Neuman for simulating, living organims, many studies have been carried off about cellular automata both on a theoretical and practical level (especially in pa~llel computing). Arthur Burks has proposed r~ ,11] a model of growing automata inspired [3,the two-dimensional cellular automata . of , " model is dedicated to the computer Th~Slce but it contains a very interesting ~~le~Ssion about the measure of the comp1e~sc of an automaton. In fact, cell ular auxlt;ata seems to be inadequate to describe torr synaptogenesis (even schematically) thE;ause of the spatial uniformity of its beSnective organization. We should deal with ~Ol at another 1eve 1, a phys i ca 1 cell bei ng 1 t ]resented by a great number of the cell s rerthe automaton. of Jt ten years ago, M.J. Apter [2] deAbol Jped a model of a 'self-constructing ve1r, which will be used as a starting-point net,ow. The basic idea of the model is the ~el110Wing one: at the beginning there is ,?ust one cell, this cell can reproduce it~self, the new cell will be identical to the bld one. The programmation of the cell is made up of a directed graph (printed for commodity inside the cell). This graph contains several categories of edges and vertices. Pulses (signals) move along edges (channels) and when a pulse enters a special node (reproduction node labe1ed by the letter R) the whole cell reproduces itself (a new cell is produced with an identical network of each cell is connected to others. Hence there is no di fference between internOl pulses and external pulses from one '
-~1-
pulse
I
®----rl) ~---E9The system is fully deterministic. The final pattern only depends upon the net of the initial cell. As there is no external information (input), we can say that we have a self-constructing network. The position of a dauthter-ce11 is entirely determined by the position of its mother-cell: in the same way the connection of the daughter cell are necessary carried off with the mother cell. In Apter's model position and connection of a cell is subordinated to its mother cell, hence depends on the reproduction process; this constitutes a practical handicap and is not justified from a neurophysiological stand point. Apter then has to improve this model, especially by assuming the existence of an internal clock capable of synchronizing some operations. Some nodes of the net are activated at precise moments; in keeping with the existence of
Model of the synaptogenesis
maturity period which has been experimentaly proved in the development of the central nervous system. It increases highly the expanding possibilities of the network. The author introduces the notions of 'input' and 'output' nodes allowing a change in the wiring of the network different from that consisting in reproducing a cell. One car: justify this method (or at least its principle) by neurophysiological experiments in which chemioaffinity seems to play a prominent part in synaptogenesis. A certain neuron (say A) sends out a specific chemical substance ; if another neuron (say B) is in a certain state and receives this signal, a new connection between A and B will be created. In fact, Apter's model is designed to prove that it is possible to make up highly complex systems from a set of relatively simple instructions. The fact that all the cells have exactly the same structure is a tool for simplifying the construction of the network. Any cell of the net contains the set of instructions necessary to produce the whole development of the system. We are now going to present our model of an evolving network.
stochastic graph grammars
Webs grammar). ~, stands for the set of messenger states (states present in the transition function of the automata). d) SC stands for the oriented and labelled graph nwhose vertices are the cells (at step n) labelled with their states and edges created by the ~rammar. e) If A is a node of SC , ~ + (A) stands for the application ~ + N dRfinRd by ~ +(A) (m) = the number of nodes B of SC , sUch as (B,A) is an edge of SC and suchnas the state of B is n. n In the same way 14 -(A) with the mode B', such as (A,B') isnan edge of SC . The couple [M +(A) , f'" -(/I)] thUs expresses both the presgnce and ~he number of the messages carried (at phasexn) by the neighbouring cells of cell A. f) The transition function is defined by: M M 4> : [. x N x N x 'j x n + f 4>
(A, [/4n +(A), Mn- (A) ], i, w) £
t
M
1I F
In the case ~ = N, i will take for value the age of A, i.e. the number of steps gone by since A was produced (either!tn the initial CS or by Webs grammar). The ECS is defined by :
11. PROPOSED MODEL ECS = We propose a new model called evolving cellular system (ECS) which is composed of an initial cellular system (CS) and of a composed webs grammar. (See Annex). Each cell of the initial CS is equiped with a stochastic automaton that determines at each step the label of this cell (we take as new label the name of the actual state of the automaton). Webs grammar acts between two iterations of the stochastic automaton and modifies the structure of the CS (for instance: connections, number of cells, etc ... ). These two aspects of the model play very di fferent parts. The CS represents a certain physical reality and stochastic automata have to mirror this reality. The grammar is just a tool to describe the set of e vents one can obse rve in the CS. The derivation rules of such a grammar have nevertheless to take into account natural processes. By that very fact, the application of a rule depends only on the state of each cell and on the connective organization. With rega rd to each cell, the t rans iti ons from one state to another may depnd on the state of the neighbouring cells, as we shall see below. Therefore we have to deal with two different and complementary processes for modi fying the structure of an ECS. We are going now to present several definitions : a) (Il,p) is a probabilised space that models the stochastic aspect of the automaton b) ~ is an input set; we shall particul arly deal with the case ~ = N (we call it clock-input). c) E = F U M is the set of the states of a cell. F stands for the set of functional states (states present as label in
173
[t,
';1, (Il ,p),
4> ,
CS o '
G)
in which CS is the initial graph (with the initial sta£e.as label) and G a Webs grammar whose lavels are taken from the set F (FS
E)·
I f ~ = Nand 4> defi neci as shown above, the system is said 'with internal clocks'. If ~1 = 0 it is said to be 'without communication' (by convention, if M = 0, 4> : E x I x n + E) . A state eo £ F is chosen as a reference state and we set that any newly created cell is in the state e . A special convent~on concerning the input of a cell created by the grammar increases the power of the ECS. This convention has it that at the moment it is created,a cell does not have the age 0 (conversely to initial cells) but it has the age of its mother cell. The mother cell of a new cell is only defined in the following case: the grammar includes a derivation rule a + 8 with more nodes in 8 than in a, and a must be reduced to single node, and this cell (say A) is called the mother cell of all the cells belonging to 8 different from the image of A. For instance: the following rule a
CJ In the first case we can say that the cell with label x is the mother of the cell carrying the label y.
174
M. Milgram
E = { (X,d) I (X,d) an edge in the host web} U = { (d,X) I (d,X) an edge in the host web} It is a rule of cell division; the cell carrying the label 'd' is the mother of the created cell .
a
DJ
..
3)
. D
In the second case there is no mother cell of the cell carrying the label z. We are going to give a very simple example of ECS with internal clock and without communication I = N E = F = {e ,d,m,e,s,r} M = 0 The follow~ng transition graph pictures the transition probabilities between states that are different from o. The probabilities are expressed below in regard with the age n of the cell. ~~~f!1el~
:
un = Prob
{ ~ (eo,n,w)
d
d}
s
..r=:l ~
e
a
E = all the edges of the host web are kept. It is a connection rule (s = source, e = entry) . We have achieved a simulation (i.e . a realization) of this cellular system with an initial CS made up of a single cell (of age 0) that cell being in a state of rest (i .e. state eo).
m
Wn
Nom
: 0
2 3
4
5
6 7 8
:e o d d d B e0 e s C eo e e D
d s r e
d s r s e
d
9
10
r r r r r r
r r r r r r
-----:----------------------------------A
n=O n>l
un
\6
wn
1 - v n
Yn
1/ 2
rn =
!i/2
\In
_!
vn -
10
n~ 5
n;.6
x n
1 - un
tn
1/ 2
\In
zn = 1 - r n The grammar contains 3 productions 1)
a
..
D
/3=cf>
(s is the graph without nodes). This Droduction mirrors the vanishing of a cell. It supposes that the empty Web exists (neither nodes nor edges) . The embedding of the empty web (0) in the host web is immediate (all the edges such as (x,m) and (m, x) are deleted) . 2)
eo
F G
r r r r r r
Each line includes the name of a cell followed by the sequence of the states taken by this cell. The final graph (all the cells are in the state r) is the following one :
n;; 5 n?6
D
0
E
r r s r
m r r r r
~.
G
Let us notice that a deterministic ECS (i.e. with a trivial probabi1ized space (n,p) reduced to a single element) that would give a similar result with the same type of webs grammar would require a great number of states (several dozens) while ours possesses only 5 of them. nn the other hand ifwe try to lower the number of states by allowing a communication between cells, the problem put by the synthesis of such an automaton is very comple x but can be approached with original methods. I I 1. DI SCUSS ION We aim at studying evolving systems of interconnected cells. The evolution rules of our system are included in a stochastic auto~aton ( a sort of genetic program) which contains a special kind of input: either they are in relation with the
Model of the synaptogenesis
age of the cell or they are messages received from neighbouring cells. Until now we have made a few prototypes of such systems work and the final note of paragraph II reports some of these remarks. We try to check general ,.hypothesis about the fonctioning and the evolution of such systems. For instance: is it possible to generate (almost surely) an element belonging to a class of graphs choosen in advance by the means of a ECS with a bounded size? Such questions have already been put by several authors about other processes generating graphs (generally in a deterministic way) [4]. The relations between the complexity of a graph and that of the automaton that generates it must certainly be studied further. Two approaches seem to be possible : a) by the use of simulating b) by achieving a consistent theory of the ECS Experience only will allow us to choose between the 2 terms of this alternative. Acknowledgments: I am pleased to acknowledge the considerable assistance of t1rs Cindy Moreau and Mr Jean-Jacques Levive for the translation of this paper. APPENDIX: Webs grammar [1,7] A weh is a directed finite graph with labeled vertices. A web grammar G is a four ~· tuple G = (V N, VT, P, S) where VN is a set of nonterminals, VT of terminals, S a set 'initial' webs, a set of web productions or rewriting A web production (or rule) is defined Cl
....
a set and P rules. as :
S (E)
where Cl and S are webs and E is an embedding of S. More precisely E consists of a set of logical functions which specify wether or not each vertex of the host web (which does not belong to S) is connected to each vertex of s. ~~~~e!~ : G = (VN' VT, P, S) V N
= { A}
V T
and P : {(p,a)
I
{(p,A) (3) :
)
,~
)
~}
•
A (p,A) an edge in the host web}
~
(2) ~
E
= {:
S
{a}
•a
(1) •a
E
=
•A I
(p,A) an edge in the hos t web}
•
a
E is the same as in (1). The langage \L(G) generated by a web grammar G is the set of all webs that we can derive from the initial set S by successively applying productions.
17.
stochastic graph grammars
In our example A
~--~·A--~-
-- ~A_ ~o_ A
o~~o~ One can easily check that the langage generated by this web grammar consists of all directed trees which have least elements. REFERENCES [1] K.S. FU - Syntactic methods in pattern recoqnition Academic Press, NY, 1974 [2] ~1.J . APTER - The genesis of neural patterns Advances in cyhernetics and systems research, GB, 1972, J., 252-268 [3]A.W . BURKS Ed. - Essays on cellular automata Univ. of Illinois Press, Urbana, Illinois,
1970 [4] A.
LINnEN~AYER, G. ROZENBERG Ed. - Automata, languages, development North Holland Publishing Company, 1976 [5] JP CHANGEUX - A. DANCHIN - Selective stabilization of developing synapses as a mechanism for the s~ecification of neuronal networks . Nature, vol 264, December 23/30, 1976 [6] J.P. CHANGEUX - P. COURREGE - A. DANCHIN A theory of the epignesis of neuronal networks by selective stabilization of synapses Proc. Nat. Acad. Sci. USA, vol 70, N°lO, 2974-2978, Oct 197~. [7] N. ABE - M. MOZIMBTO - J. TOYODA - K. TAN AKA - Web grammars and several graphs Journal of Computer and System sciences, 7,
37-65, 1973.
[8] S. WINOGRAD - J . D. COWAN - Computation in presence of noise t1IT Press '10nograph, 1963 [9] H. VON FOERSTER - On self-organizing systems and their environments In self-organizing systems, Yovitz, Cameron Ed., Pergamon Press, ~1-50, 1960 [10] H. ATLAN - L'organisation biologique et la theorie de 1 'information Hennann, Paris, 1972 [11] A. BURKS - Computation, behavior and structure in fixed and growing automata In Self-Organizing Systems, Yovitz and Cameron Ed., Pergamon Press, 282-:~09, 1960.