The Dynamics of Idealized Brain Neuron on Uncorrelated Configuration Model

1. The Dynamics of Idealized Brain Neuron on Uncorrelated Configuration Model Kyoung Eun Lee, Jae Woo Lee Inha Univ.

2. Motivation • How will the dynamics of neuron function be turned according to different network? • How will the critical property be changed as varying degree exponent, provided neural network structure is scale-free ? • How memory effect is going to affect the universality ? • What kind of the critical exponent of SOC systems can be defined and measured on scale-free network ? • We assume the neural network is composed of uncorrelated configuration model for convenience of analysis

3. 4 5 8 1 3 2 6 7 10 9 Evolution Model Rule • Uncorrelated configuration model generate with N node. • N neurons are arranged on UCM network • A random barrier Bi, equally distributed between 0 and 1, is assigned to each neurons. • At each time step, synaptic neuron is firing by locating the site with the lowest barrier Bmin by assigning a new random number to that site, and changing synaptic nearest neighbor neuron by assigning new random numbers to those sites too. • Preprocess is iterated until the system is reached the stationary state. • In stationary state, avalanche dynamics is occurred same as branching process. Generation of uncorrelated configuration model

4. 0.45 0.27 0.72 0.02 0.79 0.31 0.84 0.96 0.43 0.18 Evolution Model Rule • Uncorrelated configuration model generate with N node. • N neurons are arranged on UCM network • A random barrier Bi, equally distributed between 0 and 1, is assigned to each neurons. • At each time step, synaptic neuron is firing by locating the site with the lowest barrier Bmin by assigning a new random number to that site, and changing synaptic nearest neighbor neuron by assigning new random numbers to those sites too. • Preprocess is iterated until the system is reached the stationary state. • In stationary state, avalanche dynamics is occurred same as branching process. Generate uniform random number at each node.

5. 0.45 0.20 0.31 0.01 0.25 0.86 0.61 0.96 0.43 0.59 Evolution Model Rule • Uncorrelated configuration model generate with N node. • N neurons are arranged on UCM network • A random barrier Bi, equally distributed between 0 and 1, is assigned to each neurons. • At each time step, synaptic neuron is firing by locating the site with the lowest barrier Bmin by assigning a new random number to that site, and changing synaptic nearest neighbor neuron by assigning new random numbers to those sites too. • Preprocess is iterated until the system is reached the stationary state. • In stationary state, avalanche dynamics is occurred same as branching process. Update lowest and linked nodes to uniform random number.

6. 0.45 0.20 0.12 0.01 0.25 0.86 0.61 0.96 0.43 0.59 Evolution Model Rule • Uncorrelated configuration model generate with N node. • N neurons are arranged on UCM network • A random barrier Bi, equally distributed between 0 and 1, is assigned to each neurons. • At each time step, synaptic neuron is firing by locating the site with the lowest barrier Bmin by assigning a new random number to that site, and changing synaptic nearest neighbor neuron by assigning new random numbers to those sites too. • Preprocess is iterated until the system is reached the stationary state. • In stationary state, avalanche dynamics is occurred same as branching process. Just update the lowest neuron elapsed refractory time !

7. Definition • Self Organized Criticality:power law decay of temporal and spatial quantities, without fine-tuning of parameters. • Barrier:its instantaneous probability of releasing a spike, which is the measurement of the instability of the neuron. (low-barrier neurons are easy to fire and high-barrier neurons are difficult to fire). • Barrier Landscape:presynaptic and postsynaptic neurons that transmitted excitatory signals making the corresonding neuron unstable. • Punctuated Equilibrium:many neurons lead to the firing and new emission of a spike emerges interrupting quiet periods of apparent equilibrium, known as stasis. • Avalanche, S:elpased time or interval which fire between larger lowest barrier than auxiliary parameter B0 • Avalanche size distribution, P(S): • First Return Time Distribution, PFIRST(t):the distribution of “hole” sizes. or intervals, separating subsequent return points of active neuron to a given point in space on fractal pattern. • All Return Time Distribution, PALL(t):the probability for the active neuron at time t to revisit a site that was visited at time 0.

8. Fig 2. Average nearest-neighbor degree of vertices of degree Fig 1,2,3 is for the UCM algorithm with different degree exponents = 2, 2.5, 3, 3.5, 4, 4.5, 5 Network size is Fig 3. Average clustering coefficient Data1 Uncorrelated Configuration Model Fig 1. Degree distribution P(k) vs. k

9. Fig 4. Mininum neuron function barriers probability distribution P(B) vs. Bmin with different network degree exponents, . System size : , Refractory time : Tr=1. Fig 5. Probability distribution of the mininum barriers Bmin with different refractory times Tr System size : Degree exponent : Data2 Lowest Barrier Probability Distribution

10. Fig 7. Dependence of the exponent as a function of network degree exponents, System size : , Refractory time : Tr = 1. Fig 6. avalanche size distribution, P(B0) vs. B0 as . System size : , Refractory time : Tr = 1. Data3 Avalanche Size Distribution with Different

11. Fig 8. avalanche size distribution, P(B0) vs. B0 with different Tr . System size : , Network degree exponent : = 3. Fig 9. Dependence of the exponent as a function of the refractory time, Tr System size : , Network degree exponent : = 3. Data4 Avalanche Size Distribution with Different Tr

12. Fig 10. First return time distribution, P(t) vs. t at = 3, Tr = 1 . System size : Fig 11. All return time distribution, P(t) vs. t at = 3, Tr = 1 . System size : Data5 Return Time Distribution

13. Summary • We study neuron model on uncorrelate random scale free network. • The threshold of barrier is varying as the degree exponent, of the network, but is not changed as refractory time Tr. • While Tr = 1 is fixed and is turned from 2 to 5, the critical basic exponent, is increased until = 3 after that, is saturated to mean field value 2/3. • Inversely, at fixed = 3, the exponent is constant until Tr = 5 in errorbar as Tr is varying bottom 0 to top 10. • First and all return time distribution show the crossover having two region of power law behavior and curve.

14. Future Work • In future work, we will investigate the property of return time distribution. • We will search the transition points of network(small world and scale-free) that the memory effect of the evolution keep going robust. • We will study the neuron dynamics in brain on directed (neural) networks. • We will research neuron function for other kind of the memory effect.

15. Thank you for attention !!