Vertex Routing Models and Polyhomeostatic Optimization Claudius Gros Institute for Theoretical Physics Goethe University Frankfurt, Germany 1
topics in complex system theory Vertex Routing Models modelling conserved information flow [Markovic & Gros, NJP 9] Polyhomeostatic Optimization a new paradigm for adaptive dynamical systems [Markovic & Gros, PRL 1] 2
vertex routing models motivations criticality in dynamical systems information routing in networks cognitive processing via transient-state dynamics criticality in dynamical systems conserved quantity polynomial scaling? K = 2 Kauffman network 3
random boolean networks Kauffman networks / NK-networks N: network size K: in-connectivity random boolean functions fixpoints and cycles 1 AND 1 1 OR OR 2 3 OR 1 2 3 1 1 1 1 1 1 1 OR 1 3 2 1 1 1 1 1 1 1 AND 2 3 1 1 1 1 1 1 11 1 11 11 111 here: N = 3, K = 2 [Luque & Sole, ] 4
phase transitions in boolean networks 1 p=.9 8 CHAOS p=.79 6 p=.6 K 4 2 ORDER K: in-connectivity p: magnetization.5.6.7.8.9 1. p [Luque & Sole, ] 5
life at the edge of chaos competition: daily survival evolutionary fitness 1 p=.9 gene regulation networks 8 CHAOS p=.79 basis of all living K 6 p=.6 4 2 ORDER.5.6.7.8.9 1. p frozen (regular) phase deterministic dynamics good for daily survival bad for evolutionary adaption chaotic phase irregular dynamics bad for daily survival good for evolutionary adaption [Kauffman 69] 6
critical boolean networks scaling at criticality (K = 2)? number of attractors/cycles gene regulations networks Kauffman 69: N cell differentiation Samuelsson & Troein 3: > O(N p ) (any p)... and for other critical dynamical systems? 11 11 111 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 111 11 11 111 11 11 11 11 111 11 11 11 11 11 11 11 111 11 11 diffusive 11 11 11 11 conserved 7
information routing vs. transmision information transmission vertex vertex phase space: number of vertices information routing link (incomming) link (outgoing) phase space: number of directed links 11 111 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 111 11 11 111 11 11 11 11 111 11 11 11 11 11 11 11 11 111 11 11 transmission 11 11 11 11 11 routing 8
routing dynamics random routing tables at every vertex quenched dynamics (fixed routing tables) 1 2 (1) (2) (4)... (2) (3) (4)... information centrality overlapping cyclic attractors 4 3 number of attractors passing through a given vertex 9
information centrality numerical simulations fully connected graphs I(c,N).5.4.3.2.1 N=1 N=3 N=5 N=7.6.5.4.3.2.1.5.1.15.2 1/N 1 2 3 4 c 5 [Markovic & Gros, NJP 9] democratic distribution of the information centrality 1
cycle-length distribution numerical simulations fully connected graphs N l (L,N).25.2.15.1 N=1 N=3 N=5 N=7 µ a 15 1 5 1 2 3 4 5 6 7 N.5 5 1 15 L 2 25 3 median µ a (N) (half above/below) µ a (N) N 11
scaling of mean cycle length analytical & numerical fully connected graphs 1 σ qm ~N.95 mean ~N.81 median ~N.51 1 1 1 1 N [Schuelein, Markovic & Gros, in prep] non-trivial exponent L N.81 L : average cycle length 12
criticality in vertex routing models non-trivial distribution of (cylic) attractors mean median N.81 N σ mean N.95 N.51 = N.3.81 = N.14 N l (L,N).25.2.15.1.5 N=1 N=3 N=5 N=7 µ a 15 1 5 1 2 3 4 5 6 7 N fat tails 5 1 15 L 2 25 3 scale invariance fully connected graphs: scale-invariant Erdös-Rényi graphs: work in progrress 13
criticality in complex systems thermodynamic systems 2D Ising frozen critical chaotic critical thermodynamic systems are scale invariant and critical dynamical systems? NK-networks no? Ω 2 N vertex routing models yes? Ω N(N 1) 14
polyhomeostatic optimization homeostasis» keep in balance «a single scalar quantity... blood-sugar level hormonal levels body temperature... airplane velocity furnace temperature... polyhomeostasis multiple scalar quantities» keep in relative balance «15
allocation problems time allocation individual target distribution functions e.g. 8% working 2% socializing polyhomeostatic games goal achieve target distribution function 16
firing-rate distributions allocation of neural activities firing rate target» maximal information transmission «time Shannon (information) entropy H[p] = dy p(y)log p(y) firing-rate distribution p(y), dy p(y)=1 17
maximal information distribution maximal Shannon entropy H[p] no contraints p(y) const. given mean p µ (y) exp( y/µ), µ= y p(y)dy energy constraints target firing-rate distribution p µ (y) (polyhomeostasis) Kullback-Leibler divergence D(p, p µ ) = p(y) log ( ) p(y) dy p µ (y) asymmetric measure for the distance of two probability distribution functions 18
intrinsic plasticity distribution adaption of internal neural parameters input input strength distributions p(y) p µ (y) neural firing rate output via non-linear neural transfer function 19
stochastic adaption minimization of Kullback-Leibler divergence D a,b (p, p µ ) = p(y) log ( ) p(y) dy y(t) = p µ (y) 1 e ax(t 1) b + 1 rate-encoding neurons gain a threshold b/a output sigmoidal transfer function stochastic adaption rules input a 1/a+x ( 1 (2+λ)y+λy 2) b 1 (2+λ)y+λy 2 [Triesch, 5] 2
feed-forward polyhomeostasis.4.3 p(x).2.1 y p(y).2.1 target, µ=.28 one neuron -1 -.5.5 1 input x.5 1 neural firing y x a, b p(x) : given input distribution p(y) : output distribution 21
autapse: self-coupled neuron parameters, output 3 2 1 a(t)-4 b(t) y(t) x a, b y 1 2 3 time [Markovic & Gros, PRL 1] polyhomeostatic optimization induces continuous, self-contained neural activity limiting cycle 22
network of polyhomeostatic neurons x i (t) = j i w i j y j (t) w i j =±1/ N 1 randomly A 1 average target activities µ =.28.15 y(t) B y(t).5 49 495 5 1.8.6.4.2 5 55 6 self-organized chaos spontaneous intermittent bursting t N = 1 [Markovic & Gros, PRL 1] 23
polyhomeostatic optimization distribution of averaged neural activities p(y).2.1 target, µ=.5 network, N=1 p(y).4.3.2 [Markovic & Gros, PRL 1] target, µ=.28 network, N=1.1.5 1 firing rate y.5 1 firing rate y polyhomeostatic adaption dynamical system with local adaption rules adapting time-averaged statistics of local activities non-trivial phase diagram 24
phase diagram magnitude of average Kullback-Leibler divergence D fraction of excitatory links target mean activity D(intermittent bursting) > D(chaotic phase) N = 1 K = 1 25
intermittent route to chaos 1 y i (t).5 13 a i (t) 12 11 4 λ(t) 2 N = 1, µ=.1, fully connected activity: transient attractors intermittent bursting internal parameters: polyhomeostatic adaption towards threshold Lyapunov exonent (global, maximal) 1 2 3 4 time intermittency ˆ= punctuated equilibrium (evolution) 26
concepts and models in complex system theory complex system theory still an emergent field many models and paradigms yet to be formulated vertex routing models critical dynamical network democratic information centrality... polyhomeostatic optimization neural network / game theory / allocation problems non-trivial autonomous dynamics... 27
graduate level textbook Information theory and complexity Phase transitions and self-organized criticality Life at the edge of chaos and punctuated equilibrium Cognitive system theory and diffusive emotional control second edition fall 21 28