Modeling Neural Processes in Lindenmayer Systems

Modeling Neural Processes in Lindenmayer Systems Carlos Martín-Vide and Tseren-Onolt Ishdorj Research Group on Mathematical Linguistics Rovira i Virgili University Pl. Imperial Tàrraco 1, 43005 Tarragona, Spain carlos.martin@urv.net tserenonolt.ishdorj@estudiants.urv.es Abstract. Computing in nature as is the case with the human brain is an emerging research area in theoretical computer science. The present paper s aim is to explore biological neural cell processes of interest and to model them with foundational concepts of computer science. We have started by discovering and studying certain primitive symbolic neural operations of neuron functions, and we have formalized them with Lindenmayer (L) systems. 1 Introduction So far, the investigation of computing and learning in neural systems, or neural computation, with a good number of theoretical contributions, has been dominated by approaches from theoretical physics, information theory, and statistics [5]. Our research interest in biological neural systems is mainly focused on exploring symbolic models of neural systems through the foundations of computer science. More precisely, we attempt to explore the architecture of biological neural systems by modeling and simulating neural processes (as realistic ones as possible) with existing language-theoretic devices and bio-inspired computing techniques. We have investigated the modeling of neural processes studying, for the moment, the behavior of a single neuron concerning both its structure and some functions, as in [1], [2], [7], [8], etc. In the present paper, we explore the potentialities of Lindenmayer (L) systems for the study of neural processes, particularly impulse transmission and propagation through myelinated axon and axonal terminal tree of the neuron. We show that PD0L language sequences and length sequences of PD0L languages are thus generated. The common features shared by L systems and neural systems, both of them having distributed architectures and working in a massively parallel manner, are a good starting point for the designing of symbolic models.

2 Carlos Martín-Vide and Tseren-Onolt Ishdorj 2 Preliminaries We briefly mention next some notions, notations, and results from L systems as well as some biological neural phenomena that will be used throughout the paper. 2.1 Biological Neural Systems The basic transmitting unit in the nervous system is brain cells, so called neurons [3], [6]. The neuron is not one homogeneous integrative unit but is (potentially) divided in many sub-integrative units, each one with the ability of mediating a local synaptic output to another cell or a local electro-tonic output to another part of the same cell. Neurons are considered to consist of three main parts: a soma, the main part of the cell where the genetic material is present and life functions take place; a dendrite tree, the branches of the cell from where the impulses come in; and an axon, the branch of the neuron over which the impulse (or signal) is propagated. The branches present at the end of the axons are called terminal trees. An axon can be provided by a structure composed by special sheaths. These sheaths are involved in molecular and structural modifications of axons needed to propagate impulse signals rapidly over long distances. The impulse in effect jumps from node to node, and this form of propagation is therefore called saltatory conduction. There is a gap between neighboring myelinated regions that is known as the node of Ranvier, which contains a high density of voltagegated Na + channels for impulse generation. When the transmitting impulses reach the node of Ranvier or junction nodes of dendrite and terminal trees, or the end bulbs of the trees, it causes the change in polarization of the membrane. The change in potential can be excitatory (moving the potential toward the threshold) or inhibitory (moving the potential away from the threshold). The complexity revealed by modern research has widened with several new concepts. One of them is that the postsynaptic terminal may send retrograde signals to the presynaptic terminal. Because of this, the synapse can be viewed as having a bidirectional nature. More details about neural biology can be found in [11]. 2.2 Lindenmayer Systems In 1968, A. Lindenmayer introduced a formalism for modeling and simulating the development of multicellular organisms [4], subsequently named L systems. This formalism was closely related to the theory of automata and formal languages, and immediately attracted the attention of computer scientists [10]. The development of the theory of L systems was followed by its application to the modeling of plants. After a vigorous initial research period, some of the resulting language families, notably the families of D0L, 0L, DT0L, E0L and ET0L languages, emerged as fundamental ones. Indeed, nowadays the fundamental L families constitute a

Modeling Neural Processes in Lindenmayer Systems 3 testing ground similar to the Chomsky hierarchy when new devices (grammars, automata, etc.) and new phenomena are investigated in language theory. We recall here some formal definitions and classes of L systems. Definition 1. A finite substitution σ over an alphabet Σ is a mapping of Σ into the set of all finite nonempty languages (over an alphabet ) defined as follows. For each letter a Σ, σ(a) is a finite nonempty language, σ(λ) = λ and, for all words w 1,w 2 Σ, σ(w 1 w 2 ) = σ(w 1 )σ(w 2 ). If none of the languages σ(a), a Σ, contains the empty word, the substitution σ is referred to as λ - free or nonerasing. If each σ(a) consists of a single word, σ is called a morphism. We speak also of nonerasing and letter-to-letter morphisms. Definition 2. A 0L system is a triple G = (Σ,σ,w 0 ), where Σ is an alphabet, σ is a finite substitution on Σ, and w 0 (referred to as the axiom) is a word over Σ. A 0L system is propagating, or a P0L system, if σ is nonerasing. The 0L system G generates the languages L(G) = {w 0 } σ(w 0 ) σ(σ(w 0 )) = i 0 σi (w 0 ). Definition 3. A 0L system G = (Σ,σ,w 0 ) is deterministic, or a D0L system, if σ is a morphism. Definition 4. Let G = (Σ,h,w 0 ) be a D0L system (we use the notation h to indicate that we are dealing with a morphism). The system G generates its language L(G) in a specific order, as a sequence: w 0,w 1 = h(w 0 ),w 2 = h(w 1 ) = h 2 (w 0 ),w 3,. We denote the sequence by S(G). Thus, in connection with a D0L system G, we speak of its language L(G) and its sequence S(G). D0L systems are propagating, that is PD0L systems, if σ is nonerasing. Definition 5. Given a D0L system G = (Σ,h,w 0 ), the function f G : N N defined by f G (k) = h k (w 0 ),n 0 (1) is called the growth function of G, and the sequence F G,k = h k (w 0 ),k = 0,1,2, (2) is called its growth sequence. Functions of the form (1) are called D0L growth functions (resp. PD0L growth functions if G is a PD0L system). Number sequences of the form (2) are called D0L (resp. PD0L) length sequences. 3 L systems in modeling neural cell processes Traditionally, Lindenmayer systems are grammatical models of the development of multi-cellular organisms, which consider cells as basic atomic objects. Therefore, in Lindenmayer systems, the cells are identified by symbols. In the current section, we try to focus on zooming a neural cell. We are interested in the cell structure, and in the chemical operations that are present in the cell. We are also focused on modeling and simulating processes of the neuron in L systems. The chemicals inside cells, not cells themselves, will be atomic entities in our

4 Carlos Martín-Vide and Tseren-Onolt Ishdorj model. We consider neural cell processes - impulse transmission and propagation through myelinated axon and axonal terminal tree of the neuron - as our domain of interest and show that essential aspects of impulse transmission can be viewed as parallel rewriting processes. 3.1 Impulse transmission of axon and D0L systems The neural cell s axon structure can be mathematically represented by a linear undirected graph with n nodes labeled injectively by 1,2,,n from the ancestor node until the end node in ascending order, as illustrated in Figure 1. Let us denote an impulse by a i, the subscript i indicating which impulse sends/reaches from/into i-th node of the axon. The excite impulse a 1 is transmitted from the ancestor node with label 1 into the target node with label 2 while the impulse evolves to a 2, then it replicates into two distinguished impulses a 1 and a 3. Those impulses are transmitted to the adjacent nodes 1 and 3, and so on, the impulses being spreaded through the axon by jumping from node to node. 1 2 3 4 Fig. 1 Myelinated Axon Illustration Sending the impulse a 1 from node 1 to node 2, and sending back the impulse a n from node n to node n 1, are described by the morphisms h(a 1 ) = a 2 and h(a n ) = a n 1, respectively. The transmission of impulses from internal nodes is described as h(a i ) = a i 1 a i+1, meaning that the impulse a i replicates into a i 1 and a i+1 at the node i, and they are transmitted to the adjacent nodes i 1 and i + 1, respectively. We send a unique exciting impulse a 1 from the ancestor node into the axon with i number of nodes. We are interested in the properties held by the firing axon as impulse propagation, spread, transmission through long distance (nodes), paying special attention to sequences of PD0L. Let us consider a sequence of PD0L systems G i = (Σ i,h i,a 1 ), with Σ i = {a 1,,a i }, each of G i having morphisms h i (a 1 ) = a 2, h i (a k ) = a k 1 a k+1, h i (a i ) = a i 1, 2 k i 1, 1 i n, and the same axiom a 1. The subscript i indicates the number of nodes of the axon. Thus, G 3 = ({a 1 a 2 a 3 },h 3,a 1 ). The first few words in the sequence S(G 3 ) are a 1,a 2,a 1 a 3,a 2 a 2,a 1 a 3 a 1 a 3,a 2 a 2 a 2 a 2,a 1 a 3 a 1 a 3 a 1 a 3 a 1 a 3,a 2 a 2 a 2 a 2,. Hence, the length sequence is 1,1,2,2,4,4,8,8,. On the other hand, G 4 = ({a 1 a 2 a 3 a 4 },h 4,a 1 ). The sequence S(G 4 ) begins with words a 1,a 2,a 1 a 3,a 2 a 2 a 4,a 1 a 3 a 1 a 3 a 3,a 2 a 2 a 4 a 2 a 2 a 4 a 2 a 4,a 1 a 3 a 1 a 3 a 3 a 1 a 3 a 1 a 3 a 3 a 1 a 3 a 3. The length sequence is the well-known Fibonacci sequence 1, 1, 2, 3, 5, 8, 13,. Proposition 1. Alphabets of each adjacent words in a sequence S(G i ) are completely different: alph(w j ) alph(w j+1 ) =, j 1.

Modeling Neural Processes in Lindenmayer Systems 5 Proposition 2. Alphabets of words in even positions and in odd positions in a sequence S(G i ) coincide, respectively: alph(w i+j ) = alph(w i+j+2 ), j 1. Between the sequences of PD0L systems G i,3 i n, the next properties hold. Proposition 3. G i is obtained from previous grammars G j,j i, by using the following iterated morphism relations: h i (a k ) = h i 1 (a k ), 1 k i 2, h i (a i 1 ) = h i 1 (a i 1 )a i, h i (a i ) = a i 1, 3 i n. Word length sequences w 0, w 1, obtained from a D0L sequence S(G) = w 0,w 1,w 2, determine in a natural way the function f such that the value f(n),n 0, is defined to be the number w n in the sequence. This function is referred to as the growth function of the D0L system G. Thus, when studying growth functions, we are not interested in the words themselves but only in their lengths. Proposition 4. The numbers of occurrences of symbols in each word in the sequence are presented in the next recurrent formula. If f Gn,k(a i ) is the number of occurrences of symbol a i in the k-th word of the sequence S(G n ), then f Gn,1(a 1 ) = 1,f Gn,1(a i ) = 0, 2 i n, f Gn,k(a 1 ) = f Gn,k 1(a 2 ), f Gn,k(a i ) = f Gn,k 1(a i 1 ) + f Gn,k 1(a i+1 ), 2 i n 1, f Gn,k(a n ) = f Gn,k(a n 1 ), k 2, n 3. Proposition 5. The sum of occurrences of each symbol in the k-th word of S(G n ), n 3, (i.e. the length of that word) is: F Gn,k = f Gn,k(a 1 ) + f Gn,k(a 2 ) + + f Gn,k(a n ). (5) Proposition 6. The k-th member F Gn,k of the length sequence S(G n ) is described by the previous two members in the general formula F Gn,1 = F Gn,2 = 1, F Gn,k = F Gn,k 1 + F Gn,k 2 + Σ n 2 i=3 f G n,k 2(a i ), w n,k = w n,k 1 + w n,k 2 + Σ n 2 i=3 f G n,k 2(a i ), n 3, k 3. It is easy to claim, by using (4), that F Gn,k = f Gn,k(a 1 ) + + f Gn,k(a n ) = f Gn,k 1(a 1 ) + 2f Gn,k 1(a 2 ) + + 2f Gn,k 1(a n 1 ) + f Gn,k 1(a n ) = F Gn,k 1 + F Gn,k 2 + Σ n 2 i=3 f G n,k 2(a i ). The next simple formulas follow from (6). If the graph (axon) has 4 nodes, the tail of (6) is always 0, and Σ n 2 i=3 f G n,k 2(a i ) = 0, then S(G 4 ) is the Fibonacci numbers. F G4,1 = F G4,2 = 1, F G4,k = F G4,k 1 + F G4,k 2, k 3 or w 4,k = w 4,k 1 + w 4,k 2, k 3. (3) (4) (6)

6 Carlos Martín-Vide and Tseren-Onolt Ishdorj 3.2 Impulse transmission of the axonal terminal tree and PD0L systems The axonal terminal tree can be mathematically represented by a tree with n levels. It is possible to specify impulse transmission operations through axonal terminal trees in L systems by parallel rewriting rules. The internal nodes of the tree are hot spots. A reached impulse at a hot spot is replicated and distributed into adjacent nodes. For instance, by rule 2 341, the impulse 2 is replicated into impulses 3, 4 and 1, and distributed to the children nodes 3 and 4, and parent node 1 of the node 2, respectively. In our formalization, from the root node of the tree, impulses are only replicated and distributed to the children nodes, however from the leaf nodes impulses are just transmitted back to the parent nodes without any replication. Let us now denote an impulse as a number i. Impulses are replicated at a node and distributed only to the connected nodes as mentioned above. Here, we consider only a binary tree structure of the axonal trees. As usual, we send a unique excite impulse 1 from the root node into the tree, thus we see the process of impulse propagation and spreading through the tree. 1 1 2 3 2 2 4 4 3 3 3 3 a) b) Fig. 2 Axonal Terminal Tree Structures Consider a PD0L system G = (Σ,µ,H,1), with alphabet Σ, axonal tree structure µ, set of rewriting rules H, and axiom 1. We consider the following examples of impulse transmission in certain types of a terminal tree. Axonal terminal binary trees, illustrated in Fig. 2, are described as bracketed strings: - Three-level trees: µ a ) 1[ 2[ 44] 3] µ b ) 1[ 2[ 33] 2[ 33] ] is a complete binary tree with 3 levels. - Four-level trees: µ c ) 1[ 2[ 4[ 55] 4[ 55] ] 3] µ d ) 1[ 2[ 4[ 66] 4[ 66] 3[ 55] ] ] is a complete binary tree with 4 levels. Corresponding rewriting rules of impulse transmission and propagation through the terminal tree structures as above are: h a ) 1 23, 2 144, 3 1, 4 2

Modeling Neural Processes in Lindenmayer Systems 7 h b ) 1 22, 2 133, 3 2 h c ) 1 23, 2 144, 3 1, 4 255, 5 4 h d ) 1 23, 2 144, 3 155, 4 266, 5 3, 6 4 G a = ({1234},µ a,h a,1). The first few words in the sequence S(G a ) are 1, 23, 1441, 232223, 14411441441441, 23222323222322232223, Hence, the length sequence is 1,2,4,6,14,20,48,68,164, The length sequence S(G a ) can be generated by the following formula: w 1 = 1,w { 2 = 2, wn 1 + w w n = n 2, if n is even, w n 1 + 2 w n 2, if n is odd, n 3. A case of generating formula for the above integer sequence is found as: Sequence: 0,1,1,2,4,6,14,20,48,68,164,232,560,792,1912,2704,6528,9232, 22288,31520,76096,107616,259808,367424,887040,1254464, Name: a(0) = 0; a(1) = 1; a(n) = a(n 1) + (3 + ( 1) n ) a(n 2)/2 Formula: G.f.: x(1 + x 2x 2 )/(1 4x 2 + 2x 4 ). Example: a(4) = a(3) + 2 a(2) = 2 + 2 = 4 Author: Olivier Gerard (ogerard(at)ext.jussieu.fr), Jun 05 2001. Let us consider a PD0L system G b = ({123},µ b,h b,1). The first few words in the sequence S(G b ) are 1,22,133133,22222222,133133133133133133133133, Hence, the length sequence is 1,2,6,8,24,32,96, The language generated by the grammar G b is L(G b ) = {1} {(22) 22n,(133) 22n+1 n 0}. The length sequence S(G b ) can be generated by the following formula: w 1 = 1,w { 2 = 2, wn 1 + w w n = n 2 = 2 n 1, if n is even, 3 w n 1 = 3 2 n 2, if n is odd, n 3. G c = ({12345},µ c,h c,1). The first few words in the sequence S(G c ) are 1,23,1441,2325525523,144114444144441441, 232552552323255255255255232552552552552325525523,. Hence, the length sequence is 1, 2, 4, 10, 18, 48,. 4 Final Remarks We have tried to show that the application of L systems to neural process as impulse transmission and propagation through an axon with n nodes and terminal trees produce PD0L languages and length sequences of languages. The growth function for impulse propagation in the axon (5) tells us that the number of impulses propagated in the axon is growing in an exponential manner,

8 Carlos Martín-Vide and Tseren-Onolt Ishdorj and this number does not depend on the number of nodes in the axon. We have seen that, depending on the number of nodes in the axon, impulse transmission generates different types of sequences of numbers, but all of them share the common formula (6). Modeling the whole information transmission process in neurons, receiving impulses in the dendrite tree, transmitting them into the soma, processing impulses in the soma, transmitting the processed impulses into the terminal tree, finally releasing the impulses into the connected junctions of neurons in order to make neural connections, all of this could be simulated with L systems. Acknowledgments. The work partially supported by the Asian Research Center, National University of Mongolia. References 1. Cavaliere, M., Ionescu, M., Ishdorj, T.-O., Inhibiting/De-inhibiting Rules in P Systems, Proceedings 5 th Workshop on Membrane Computing, Milano, Italy, 2004, 60 73, and Lecture Notes in Computer Science LNCS 3365, 224 238, Springer, Berlin Heidelberg 2005. 2. Ishdorj, T.-O., Ionescu, M., Replicative-Distribution Rules in P Systems with Active Membranes, Proc. First International Colloquium on Theoretical Aspects of Computing, Guiyang, China, September 20-24 (2004) - UNU/IIST Report No. 310, 263 278, Zhiming Liu (Ed.) Macau, and Lecture Notes in Computer Science LNCS 3407, 69 84, Springer, Berlin Heidelberg (2005). 3. Kleene, S.C., Representation of Events in Nerve Nets and Finite Automata. In Automata Studies, Princeton University Press, Princeton, NJ, 1956, 3 42. 4. Lindenmayer, A., Mathematical models for cellular interaction in development I and II. Journal of Theoretical Biology 18(1968), 280 315. 5. Maass, W., Neural Computation: A Research Topic for Theoretical Computer Science? Some Thoughts and Pointers, in [9]. 6. McCulloch, W.S., Pitts, W.H., A Logical Calculus of the Ideas Immanent in Nervous Activity. Bulletin of Mathematical Biophysics, 5(1943), 115 133. 7. Pan, L., Alhazov, A., Ishdorj, T.-O., Further Remarks on P Systems with Active Membranes, Separation, Merging and Release Rules. Soft Computing, 8(2004), 1 5. 8. Pan, L., Ishdorj, T.-O., P Systems with Active Membranes and Separation Rules. Journal of Universal Computer Science, 10, 5(2004), 630 649. 9. Păun, Gh., Rozenberg, G., Salomaa., A.,(eds), Current Trends in Theoretical Computer Science: Entering the 21 st Century, World Scientific, Singapore, 2001. 10. Rozenberg, G., Salomaa, A., The Mathematical Theory of L Systems, Academic Press, New York, 1980. 11. Shepherd, G.M., Neurobiology, Oxford University Press, Oxford, 1994.