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Yo Horikawa Kagawa University Japan

Noise-Induced Increases in the Duration of Transient Oscillations in Ring Neural Networks and Correlations in the Periods of Ring Oscillators. Yo Horikawa Kagawa University Japan. 2. 1. N. 3. 4. 5. 6. 7. 8. 1. Problem

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Yo Horikawa Kagawa University Japan

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  1. Noise-Induced Increases in the Duration of Transient Oscillations in Ring Neural Networks and Correlations in the Periods of Ring Oscillators Yo Horikawa Kagawa University Japan

  2. 2 1 N 3 4 5 6 7 8 1. Problem Ring network of neurons with unidirectional coupling with noise dxn(t)/dt = -xn(t) + f(xn-1(t)) + σxnn(t) (x0 = xN, 1 ≤ n ≤ N) f(x) = tanh(gx) E{nn(t)} = 0, E{nn(t)nn’(t’)} = δn,n’·δ(t - t’) (A1) xn: state of neuron n N: number of neurons f(x) = tanh(gx): output of neuron g: coupling gain (|g| > gth(N) ≥ 1) nn(t): spatiotemporal white noise σx: strength of noise

  3. 1. Effects of noise on the duration of transient oscillations Number of neurons is even(N = 2M). → Network is bistable. Duration T of the transient oscillations increases exponentially with the number N of neurons. T∝ exp(N) → Spatiotemporal noise of intermediate strength (σx) increases the mean duration (m(T)) of the transient oscillations. 3

  4. T1T2T3T4 ･･･ 2. Effects of noise on the periods of stable oscillations Number of neurons is odd (N = 2M + 1). → Ring oscillator (Ring of inverters (NOT gates)) The oscillations are stable. → Spatiotemporal noise causes positive correlations in a series of the half periods (a series of the widths of the pulses) of the oscillations. Power spectrum S(ω) of a series {Tj} of the half periods of the oscillations 4

  5. 2 1 2m 3 4 5 - - - + + 6 x = (+ - ･･･ + -) OR x = (- + ･･･ - +) 7 8 2. Effects of noise on the duration of transient oscillations Number of neurons is even(N = 2M). → Network is bistable. Transient oscillations: Traveling waves of two inconsistencies in the signs of the states of neurons Changes in duration of transient oscillations due to noise 5

  6. t We use excitatory coupling. Neurons are separated in two blocks of the same signs. (- - - - - -- - - + + + + + ++ + + +) (+ - - - - - -- - - + + + + + ++ + +) (+ + - - - - - -- - - + + + + + ++ +) (+ + + - - - - - -- - - + + + + + ++) (+ + + + + + + + + + + + - - + + +) (+ + + + ++ + + + ++ + + -+ + +) (+ + + + + + + + + + ++ + + + ++) Demonstration Noise changes the duration of the transient oscillations. dx1/dt = -x1 + f(xN) dxn/dt = -xn + f(xn - 1) (2 ≤ n ≤ N) (A2) xn: state of neuron n f(x) = tanh(gx): output of neuron g: coupling gain (g > 1) 6

  7. v1 v0 l(t) (+ + + + + + + + + + + + + - - - -- - - -- - -- - + + + + + + + + + + + + + +) Difference between the velocities of two blocks: v0 – v1 < 0 → Changes in the length of a smaller block: l(t) with dl(t)/dt = v0 – v1 → Duration T of the oscillation with the condition l(T) = 0 7

  8. xn -1(t) t Δt0 τdxn/dt = -xn - 1 Δt1 xn(t) t t0 t1 τdxn/dt = -xn + 1 Sigmoidal function: f(x) = tanh(gx) → Sign function: sgn(x) dxn/dt = -xn - 1 + σxnn(t) (xn -1 < 0) = -xn + 1 + σxnn(t) (xn -1 > 0) (A3) v0 v1 l(t) dl(t)/dt = v0 - v1 = 1/Δt0 - 1/Δt1 Δt1 ≈ τlog2 + log{1 - exp[-(t1 - t0)/τ]} + σ’n1(t) Δt0 ≈ τlog2 + log{1 - exp[-(t0 – t-1)/τ]} + σ’n2(t) dl(t)/dt≈ 1/(log2)2·[exp(-log2·(L - l)) - exp(-log2·l)] +σln(t) σl2 = 2σ2(tp)/(log2)4 = 3/[4(log2)4]·σx2 E{n(t)} = 0, E{n(t)n(t’)} = δ(t - t’) (A4) x(t0) ≈ - 1 8

  9. T3 T1 T0 T2 Changes in the length of a smaller block: dl(t)/dt ≈ 1/(log2)2·[exp(-log2·(L - l)) - exp(-log2·l)] +σln(t) σl2 = 2σ2(tp)/(log2)4 = 3/[4(log2)4]·σx2 ≈ 3.25 σx2 E{n(t)} = 0, E{n(t)n(t’)} = δ(t - t’) (A4) The duration T of the transient oscillations is dealt with the first passage time (FPT) problem of Eq. (A4) from l(0) = l0 to l(T) = 0 or L. 9

  10. a(l) = 1/(log2)2·[exp(-log2·(L - l)) - exp(-log2·l)] (A5) The mean m(T(l0)) of the FPT beginning from the initial block length l0 are given in the integral form. Fig. A1. Mean duration m(T(l0)) of the transient oscillations vs. SD σx of the noise. N = 40 and l0 = 15. FPT for simulation with tanh(gx) (g = 10) is multiplied by the ratio (1.24) of the durations without noise. The mean duration m(T) of the transient oscillations occurring from a fixed initial block length l0 increases in the presence of noise with intermediate strength. 10

  11. When an initial length l0 of a smaller block changes, the noise strength σx at a peak of the mean duration m(T) changes. The mean duration m(T) of the transient oscillations occurring under random initial conditions decreases monotonically with the strength σx of noise. Fig. A2. Mean duration m(T(l0)) of the transient oscillations vs. SD σx of the noise. N = 40 and l0 = 12, 13, 14, 15. 11

  12. 2 1 2m+1 3 4 5 6 - + + 7 8 + - + -- + - ++ - + - ++ - + -- + - + t 3. Effects of noise on the periods of stable oscillations Number of neurons is odd (N = 2M + 1). → Ring oscillator: Ring of inverters (NOT gates) Stable oscillations of rectangular waves Demonstration (N = 3)

  13. Variations in a series of the half periods Tj of the oscillation due to noise N = 3 ･･･ Tj -1TjTj +1 ･･･ 13

  14. xm xn -1(t) Tm xn(t) Δtm f(x) = tanh(gx) → Sign function: sgn(x) dxn/dt = -xn - 1 + σxnn(t) (xn -1 < 0) = -xn + 1 + σxnn(t) (xn -1 > 0) Condition for the stable oscillation in the absence of noise: xm = -(xm + 1)exp(-Tm) + 1 (xm > 0) Propagation time of the inconsistency per neuron: Δtm = log(xm + 1) Half period: Tm = NΔtm Maximum value of the state of neurons: xm = tanh(Tm) N = 3 14

  15. tj -1 tj Tj xn(tj) = (xn(tj -1) - 1)exp(-Tm) + 1 + σt’’nj(t) dxn(t)/dt = -xn(t) - 1 + σxnn(t) (xn -1(t) < 0) xn(tj + Δtj) = 0 Δtj xn(tj -1) The propagation time Δtj of the jth passing of the inconsistency in the presence of noise: Δtj ≈ log{1 + xm + exp[-(N - 1)m(Δt)][(N - 1)/N]T’j} + σtnj(t) ≈ Δtm - exp(-Tm)[(N - 1)/N]Tj’ + σtnj(t) Tj = tj - tj -1, Tj’ = Tj’ = Tj - Tm σt2 ≈ σx2/2·{[1 - exp(-2Δtm)] + [1 – exp(-2(N - 1)Δtm)]exp(-2Δtm)} (A6) 15

  16. Changes in the half period Tj at the location l in the network is approximated as dTj(l)/dl = d(tj - tj-1)/dl ≈ Δtj - Δtj-1 ≈ β(Tj - Tj-1) + σt(nj - nj-1) β = exp(-Tm)·(N - 1)/N, Tj(0) = Tj-1(N) Tj(0) = Tj -1(N) (nj: white noise along tj) (A7) Z transform ZT of Tj is given by dZT(l)/dl = β(1 - z-1)ZT(l) + (1 - z-1)Zn(l) ZT(0) = z-1ZT(N) (A8) 16

  17. The power spectrum S(ω) of Tj is obtained as S(ω) = E{|ZT(N)|2 z = exp(iω)} = σt2/β·{exp[2βN(1 - cos(ω))] – 1} /{1 + exp[2βN(1 - cos(ω))] - 2exp[βN(1 - cos(ω))]cos[ω - βNsin(ω)]} (A9) The power spectrum is expressed with exponentials of cos(ω) and sin(ω). When |βN| < 1, a sequence of the half periods is approximated by the first-order autoregressive (AR) process. Tj(N) = tj(N) - tj -1(N) = ≈ βN/2·(Tj -1(N) + Tj(N)) + σTnj(approximation of the integral with the trapezoidal rule) = φ1Tj -1(N) + σTnj φ1 = βN/(2 - βN) > 0, σT2 = σt2N/(1 - βN/2)2 Power spectrum: S(ω) = σT2/(1 - 2φ1cos(ω) + φ12) Autocovariance function: E{(Tj(l) - Tm)(Tj-k(l) - Tm)} = σT2/(1 - φ12)·φ1k (A10) 17

  18. ･N = 3 xm = (-1 + 51/2)/2 ≈ 0.618 Δtm = log((1 + 51/2)/2) ≈ 0.481 Tm = log(2 + 51/2) ≈ 1.44 dTj(l)/dl = β(Tj - Tj -1) + σt(nj - nj -1) β = 2/3·(-2 + 51/2) ≈ 0.157 σt2 = 2(-2 + 51/2)σx2 ≈ 0.472σx2 S(ω) = σt2/β·{exp[2βN(1 - cos(ω))] – 1} /{1 + exp[2βN(1 - cos(ω))] - 2exp[βN(1 - cos(ω))]cos[ω - βNsin(ω)]} Tj(l) = φ1Tj -1(N) + σTnj φ1 = (-1 + 51/2)/4 ≈ 0.309 σT2 = 3/4·(1 + 51/2)σx2 ≈ 2.43σx2 S(ω) = σT2/(1 - 2φ1cos(ω) + φ12) Noise causes positive correlations in a series of the half periods. Fig. A3. Power spectra S(ω) of a series Tj of the half periods in the ring oscillator of three neurons (N = 3) in the presence of noise of σx = 0.1. 18

  19. Inverter Inverter Inverter Summing Amplifier Summing Amplifier Summing Amplifier R R R ㊉ ㊉ ㊉ C C C Inverting Amplifier Inverting Amplifier Inverting Amplifier x10 x10 x10 FM Radio ･･･ Tpj -1Tpj ･･･ ･･･ T2j -2T2jT2j+1T2j -2 ･･･ Circuit experiment N = 3 Noise source: untuned FM radio, Inverter: TC4049, Amplifiers: RC4558 R = 10kΩ, C = 1μF, (time constant: CR = 10ms) Oscilloscope Small shift in a zero-level causes large negative correlations in a series of the half periods. → A series of the periods: Tpj = T2j + T2j+1 19

  20. Downsampling by a factor 2 of a series of the sums of the successive 2 half periods →a series of the periods: Tpj = T2j + T2j+1 The power spectrum S(ω; Tpj) of a series of the periods: S(ω; Tpj) = (1 + cos(ω/2))S(ω/2; Tj) + (1 + cos(π - ω/2))S(π - ω/2; Tj) (A11) S(ω; Tj) = σt2/β·{exp[2βN(1 - cos(ω))] - 1} /{1 + exp[2βN(1 - cos(ω))] - 2exp[βN(1 - cos(ω))]cos[ω - βNsin(ω)]} (A9) S(ω; Tj) = σT2/(1 - 2φ1cos(ω) + φ12) (A10) Fig. A4. Sample of a series Tpj of the periods in the ring oscillator of three neurons (N = 3) in a circuit experiment. Fig. A5. Power spectra S(ω) of a series Tpj of the periods in a circuit experiment. 20

  21. 4. Conclusion Effects of spatiotemporal noise on the oscillations in the ring networks of neurons were studied. 1. Duration of the transient oscillations in the networks of even numbers of neurons (N = 2M) → Mean duration of the transient oscillations increases owing to noise of intermediate strength. Noise-sustained oscillations 2. Variations and correlations in a series of the half periods of the stable oscillations in the networks of odd numbers of neurons (N = 2M + 1) (Ring oscillator) → Noise causes positive correlations in a series of the half periods. The power spectra are of exponential type. Similar to the interspike intervals of a spike propagating in a nerve fiber 21

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