1 Properties of the Duration of Transient Oscillations in a Ring Neural Network Yo Horikawa and Hiroyuki Kitajima Kagawa University Japan

2 1. Background Ring network of sigmoidal neuron models with inhibitory unidirectional coupling When the coupling gain: |g|> g th (N) ≥ 1, ･ Number of neurons: odd (N = 2M + 1) → Stable oscillation ･ Number of neurons: even (N = 2M) → Stable steady states (e.g. x 2k ≈ 1, x 2k-1 ≈ -1 (0 ≤ k ≤ m) τdx 1 /dt = -x 1 + f(x N ) τdx n /dt = -x n + f(x n - 1 ) (2 ≤ n ≤ N) (A1) x n : state of neuron n N: number of neurons f(x) = tanh(gx): output of neuron g: coupling gain (g < 0) τ: time constant 4 5 N

3 ･ Number of neurons is odd (N = 2M + 1). → Ring oscillator: Ring of inverters (NOT gates) Stable oscillations of rectangular waves 4 5 2M t τdx 1 /dt = -x 1 + f(x N ) τdx n /dt = -x n + f(x n - 1 ) (2 ≤ n ≤ N) (A1) x n : state of neuron n N: number of neurons f(x) = tanh(gx): output of neuron g: coupling gain (g < 0) τ: time constant xn(t)xn(t)

4 ･ Number of neurons is even (N = 2M). → Network is bistable. However, transient states from random initial values can be extremely long. Transient oscillations lasted more than a month in computer simulation with a workstation for N = M2M x = (+ - ･･･ + -) OR x = (- + ･･･ - +) Simulation N = 40, g = -10.0, τ= 1.0, x n (0) ~ N(0, ) (1 ≤ n ≤ 40) Circuit experiment n = 40, C = 10μF, R = 10kΩ (time constant: CR = 0.1s) Oscillation lasts about 45 minutes.

5 The transient oscillations are traveling waves of the boundaries of separated blocks. Scenario to the steady state The neurons are separated into two blocks in which the signs of the states of the neurons change alternately. ( ･･･ ･･･ -+-+). Two boundaries (inconsistencies) of the two blocks move in the direction of the coupling by changing the signs of the states of the neurons. ( ･･･ ･･･ -+-+) →( ･･･ ･･･ -+-+) →( ･･･ ･･･ -+-+) The velocities of the boundaries differ only slightly and continue to move for a long time. ( ･･･ ･･･ -+-+) →( ･･･ ･･･ -+-+) They finally merge and the network reaches the steady state.

6 We consider excitatory coupling instead of inhibitory coupling for simplicity. Neurons are separated in two blocks of the same signs. ( ) ( ) ( ) ( ) ( ) ( ) ( ) Demonstration τdx 1 /dt = -x 1 + f(x N ) τdx n /dt = -x n + f(x n - 1 ) (2 ≤ n ≤ N) (A2) x n : state of neuron n f(x) = tanh(gx): output of neuron g: coupling gain (g > 1) t

7 2. Velocity of the boundaries Random initial values of x n (0) (1 ≤ n ≤ N) (after short time) → Propagating two blocks ( ) Difference between the velocities of two blocks: v 0 – v 1 < 0 → Changes in the length of a smaller block: l(t) with dl(t)/dt = v 0 – v 1 → Duration T of the oscillation with the condition l(T) = 0 v1v1 v0v0 l(t)l(t) l(t): the length of a smaller block The number of neurons in the block

8 Propagation time of the boundary per neuron: Δt → Velocity of the block per neuron: v = 1/ Δt Sigmoidal function: f(x) = tanh(gx) → Sign function: sgn(x) τdx n /dt = -x n - 1 (x n -1 < 0) = -x n + 1 (x n -1 > 0) (A3) τdx n /dt = -x n + 1τdx n /dt = -x n - 1 x(t 0 ) ≈ - 1 n n

9 τdx n /dt = -x n + 1 (x n -1 > 0) x n (t 1 ) = exp(-(t 1 - t 0 )/τ)(x n (t 0 ) - 1) + 1 ≈ -2exp(-(t 1 - t 0 )/τ) + 1 τdx n /dt = -x n - 1 (x n -1 < 0) x n (t 1 +Δt 1 ) = exp(-Δt 1 /τ)(x n (t 1 ) + 1) - 1 = 0 Δt 1 =τlog{2 - 2exp[-(t 1 - t 0 )/τ]} =τlog2 + log{1 - exp[-(t 1 - t 0 )/τ]} v 1 = 1/Δt 1 = 1/[τlog{2 - 2exp[-(t 1 - t 0 )/τ]}] ≈ 1/[τlog{2 - 2exp[log2 ･ l(t)]}] v 0 = 1/Δt 0 = 1/[τlog{2 - 2exp[-(t 0 – t -1 )/τ]}] ≈ 1/[τlog{2 - 2exp[log2 ･ (N - l(t))]}] dl(t)/dt = v 0 - v 1 = 1/Δt 0 - 1/Δt 1 ≈ 1/[τlog{2 - 2exp[log2 ･ (N - l(t))]}] - 1/[τlog{2 - 2exp[log2 ･ l(t)]}] Δt ≈ τlog2 (t 1 - t 0 ) → ∞

10 dl/dt ≈ 1/[τlog{2 - 2exp[log2 ･ (N - l)]}] - 1/[τlog{2 - 2exp[log2 ･ l]}] = 1/{τ[log2 + log( (N - l) )]} - 1/{τ[log2 + log( l )]} ≈ 1/[τ(log (N - l) )] - 1/[τ(log l )] ≈ 1/τ·[1/log (N - l) /(log2) 2 - (1/log l /(log2) 2 )] = 1/[τ(log2) 2 ]·(2 -(N - l) - 2 -l ) = k(exp(-c(N - l)) - exp(-cl)) (k = 1/[τ(log2) 2 ], c = log2) (A4) The velocity of the boundary depends on an exponential of distance to the forward boundary. The difference between the velocites of two boundaries decreases exponentially with the length l and N - l of the blocks. Changes in the length of the blocks is exponentially small when the block lengths are large. n = 1 N

11 3. Duration of the transient oscillations dl/dt = k(exp(-c(N - l)) - exp(-cl)) (k = 1/[τ(log2) 2 ], c = log2) This equation is solved as exp(cl(t)) = exp(cN/2)tanh(-exp(-cN/2)ckt + arctanh(exp(cl 0 - cN/2))) (l 0 = l(0) (0 ≤ l 0 ≤ N/2)) (A5) The two blocks merge, i.e. a smaller block disappear. → The oscillation ceases. The duration T of the transient oscillations is given with l(T) = 0. T = 1/(ck)·exp(cN/2){arctanh[exp(c(l 0 - N/2))] – arctanh[exp(-cN/2)]} (A6) T 10 T 15 T 18 T 22

12 A simple form of the duration T(l 0 ) when the number N of neurons is large is obtained by letting N be infinity (N → ∞). dl/dt = -kexp(-cl) l(t) = 1/c·log(exp(cl 0 ) - ckt) T = (exp(cl 0 ) - 1))/ck = τlog2·(2 l 0 - 1) (l(T) = 0) (k = 1/[τ(log2) 2 ], c = log2, l 0 = l(0) (0 ≤ l 0 ≤ N/2)) (A7) The duration of the transient oscillations increases exponentially with the length of a smaller block, i.e. the number of neurons in the block. Fig. A1. Duration T of the transient oscillations with the initial block length l 0 (1 ≤ l 0 ≤ 19) and N = 40.

13 4. Distribution of the duration under random initial conditions Random initial states x n (0) (1 ≤ n ≤ N) of neurons ~ i. i. d. → The initial length of a smaller block is distributed uniformly in (0, N/2). l 0 ~ U(0, N/2). The probability density function h(T) of the duration T of the transient oscillations is derived with h(T) = |dT(l 0 ; N)/dl 0 | -1 /(N/2) = 2kexp(-cN/2)cosech[2(exp(-cN/2)ckT + arctanh(exp(-cN/2)))]∙2/N = 2/[τ(log2) 2 ]∙2 -N/2 cosech[2(2 -N/2 T/[τlog2] + arctanh(2 -N/2 ))]∙2/N (A8)

14 The distribution of the duration T under random initial condition. h(T) = 2/[τ(log2) 2 ]∙2 -N/2 cosech[2(2 -N/2 T/[τlog2] + arctanh(2 -N/2 ))]∙2/N (A8) A cut-off point: T c = τlog2·2 N/2 ･ T < T c h(T) ≈ k/(ckT+1)∙2/N = 1/[τ(log2) 2 (T/(τlog2) + 1)]∙2/N (0 ≤ T ≤ 1/ck·(exp(cN/2) - 1) = τlog2·(2 N/2 - 1)) (A9) The duration T is thus distributed in the form of 1/T. ･ T > T c h(T) ≈ λexp(-λT) (λ≈ 2exp(-cN/2)ck = 1/[2 N/2-1 ∙τ(log2)]) (A10) The cut-off point T c increases exponentially with the number of neurons. Proportion of the duration of the oscillations over the cut-off is inversely proportional to the number N of neurons. Pr{T < T c } ∝ 1/N → The inverse power-law distribution dominates as N increases.

15 Fig. A2. Probability density function h of the duration T of the transient oscillations in the network with the numbers N of neurons 10 (a), 20 (b), 40 (c). h ∝ 1/T

16 The mean m, the variance σ 2 and the coefficient of variation CV of the duration T is derived with Eq. (A9). m(T(N)) = 2(exp(cN/2) - 1 – cN/2)/(c 2 kN) = 2τ(2 N/ log2/2∙N)/N σ 2 (T(N)) = (exp(cN) - 4exp(cN/2) cN)/(c 3 k 2 N) - {m(T(N)} 2 CV(T(N)) = σ(T(N))/m(T(N)) ≈ (cN) 1/2 /2 = (log2) 1/2 /2∙N 1/2 (N » 1) (A11) Fig. A3. Mean m(T) of the duration of the transient oscillations vs Number N of neurons. The mean duration m(T) increases exponentially with the number of neurons.

17 5. Conclusion Properties of the transient oscillations in the ring networks of neurons with inhibitory unidirectional coupling were studied. ･ Kinematical model of the traveling waves in the networks The transient oscillations are the propagating boundaries of the blocks of the neurons at which the signs of the state of neurons are inconsistent. The kinematical equation of the propagating boundaries was derived. The difference between the velocities of the two propagating boundaries is exponentially small with the length of the blocks. ･ Exponential increases in the duration of the oscillations with the number of neurons The duration of the transient oscillations increases exponentially with the number of neurons. ･ Power-law distributions of the duration of the oscillations The distribution of the duration of the oscillations is in the form of the inverse power-law below the cut-off.

18 Transient states increasing exponentially with system size ･ Transient spatio-temporal chaos in coupled map lattices [A1] and reaction-diffusion equations [A2] ･ The length and number of cycles and transient time to them in asymmetric neural networks [A3] ･ Transient irregular firings in diluted inhibitory networks of pulse-coupled neurons [A4] ･ Transient well-controlled sequences in continuous-time Hopfield networks with Liapunov functions [A5]. These systems never reach their asymptotically stable states in a practical time when the system size is sufficiently large. Their functions, e.g. information processing in the nervous systems, may proceed in the transient states. The transient states thus play more important roles than the asymptotic states in actual systems. [8] K. Kaneko, Supertransients, spatiotemporal intermittency and stability of fully developed spatiotemporal chaos, Phys. Lett. A 149 (1990) pp [9] A. Wacker, S. Bose and E. Schöll, Transient spatio-temporal chaos in a reaction-diffusion model, Europhysics Letters 31 (1995) pp [10] U. Bastolla1 and G. Parisi, Relaxation, closing probabilities and transition from oscillatory to chaotic attractors in asymmetric neural networks, J. Phys. A 31 (1998) pp [11] R. Zillmera, R. Livib, A. Politic and A. Torcini, Desynchronized stable states in diluted neural networks, Neurocomputing 70 (2007) pp [12] J. Šíma and P. Orponen, Exponential transients in continuous-time Liapunov systems, Theoretical Computer Science 306 (2003) pp