A General Learning Rule for Network Modelingof Neuroimmune Interactome

D. Remondini1,3,4, P. Tieri1,2, S. Valensin1,2, E. Verondini1,4, C. Franceschi1,2,F. Bersani1,4, and G.C. Castellani1,3,4

1 “L.Galvani” Interdipartimental Center for Biophysics,Bioinformatics and Biocomplexity

2 Department of experimental Pathology3 DIMORFIPA

4 Physics Department and INFN,University of Bologna

daniel.remondini@unibo.it,gastone.castellani@unibo.it

Abstract. We propose a network model in which the communicationbetween its elements (cells, neurons and lymphocytes) can be estab-lished in various ways. The system evolution is driven by a set of equa-tions that encodes various degrees of competition between elements.Each element has an “internal plasticity threshold” that, by settingthe number of inputs and outputs, determines different network globaltopologies.

1 Introduction

The network description of Immune System (IS) began in the 70’s [15]. Thenetwork theory of IS was further developed over the past 30 years with sev-eral approaches[17, 8, 3]. The network approach on IS was influenced by NeuralNetwork theories and modelling that, in their original formulation, started inthe 40‘s and was developed along various lines and perspectives [12, 2]. Amongthe various learning rules we will consider one special class : the energy basedlearning rule [4], where the weight updating is obtained by minimization of a“Risk” or “Energy Function”[12, 14]. During the last years the so called “Com-plex network theory” showed several progresses in theoretical and applicativefields[22, 21]. In these models the evolution of links is obtained by various “rules”[22]. In the contemporary BioMedicine, concepts like transcriptome, interac-tome, etc are network-inspired. The effect of connection topology on immuneand neural networks is still not completely understood, and the the existingmodels are not completely characterized on the basis of the generated connec-tivity patterns [9]. One of the present challenges is a better understanding of howmicroscopic relations between and within network elements can lead to macro-scopic phenomena such as adaptation, learning and memory, and how geneticand environmental information fluxes interact. This work deals with the con-nectivity problem, which is not new in the frame of IS models [6, 20]. Among

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various neural network the so called BCM (Bienenstock, Cooper and Munro)neural network (BCM NN)[2, 14, 7, 5] shows an interesting similarity in term ofstability of “learning rule” solutions with a class of immune network models[3].The relation between immune and nervous system looks very intriguing sinceboth systems can be described, from a mathematical-systemic point of view,by similar equations based on a “network” structure [10, 15]. Recent studiessuggest that the similarity between the two systems is not purely formal [11].Among others, a surprising result was [1] that some fundamental molecules forlearning and memory in the nervous system (like CaMK II and Calcineurin)play a fundamental role also in the IS, leading to a new scenario in whichmolecules and pathways are shared by the two systems and conserved duringevolution. In this paper we present a learning rule, that can be applied to awide class of networks to model some emergent properties such as memory andlearning.

2 Model

We propose a model in which generic units (neurons, lymphocytes etc) can com-municate by links among them. Each unit is specified by its inputs, outputs andan internal parameter: a history dependent threshold. This threshold dependson the time average of incoming and outgoing signals and maps the history ofeach unit to a plasticity function. This assumption is justified by the fact that,in terms of internal activity and methabolic energy, the receiving and sendingactivities are “active” processes. The “plasticity”function Φ is responsible forthe strengthening and weakening of links between nodes. The definition of Φis choosing according to statistical considerations and mathematical simplicity,but the general results are true for a wide class of changing sign functions [14].The evolution equation are:

duij

dt= uijΦij i = j, 1 . . . , n. (1)

Φij = uij − Θij i = j, 1 . . . , n. (2)

Θij =∑

i=j∈Ωij

u2ij (3)

where uij are the link values and Ωij are appropriate subsets of the networklinks. The choice of the Ω subsets is crucial, since the development of differentnetwork structures critically depend on this choice.

3 Results

Our results are obtained by varying the number of elements (input and output)in the definition of the threshold Θ. In the case of a global threshold where all

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the units interact each other we obtain a directed network in which one unitgets all the incoming (or outgoing) links (see Fig. 1). This is an extreme caseof “unfair” link distribution. By weakening this assumption we are able to getintermediate cases in which few units get most of the links, with a “fat-tailed”connectivity degree distribution (see Fig. 3). If the competition is restricted

Fig. 1. Directed graph generated from the evolution rule with a threshold such thatone unit wins all

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Fig. 2. Directed graph where each unit has only one incoming link and randomlydistributed outgoing links

to all of the incoming (or outgoing) links for each unit, we obtain completeselectivity (only one directed link) for the input but leaving the output uncon-strained, and viceversa (see Fig. 2). We remark that all these networks are differ-ent from a random generated network, because they embody multiple structuresthat can be further analyzed and experimentally tested in terms of biologicalinterpreteation.

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Fig. 3. Directed graph where the “one unit wins all” rule has been weakened

4 Conclusion

This analysis is one of the first attempts to build a network based on a learn-ing rule that can produce different topologies. Our network is characterized bygeneric units that can be neurons, lymphocytes or generic cells. The learningrule has a direct interpretation in the development of network connectivity be-cause the history dependent threshold has a biological counterpart in biological

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phenomena such as memory induction in neurons and lymphocytes. Our nu-merical results show that we can tune the network topology, from random toscale-free-like connectivity degree distributions, by changing the links containedin the definition of the threshold Θ. We think that this is a first step towardsmore realistic neural and immune models taking into account the internal struc-ture [18, 19] of cell communication and the role of signalling molecules in theinduction of learning and memory.

Acknowledgments. This work has been supported by Italian FIRB grant,INFN FB11 grant and a Bologna University grant ( ex 60 % ).

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