1. 1 Experiments on Transient Oscillations in a Circuit of Diffusively Coupled Inverting Amplifiers Yo Horikawa Kagawa University, Japan

2. 2 1. Background Ring network of neuron models with inhibitory one- way coupling When the coupling gain g > 1 Number of neurons: odd (n = 2m+1) → Stable oscillation Number of neurons: even (n = 2m) → Stable steady state (e.g. x 2k = 1, x 2k-1 = -1 (0 ≤ k ≤ m) 4 5 n 12 6 7 8 3 τdx 1 /dt = -x 1 - f(gx n ) τdx i /dt = -x i - f(gx i-1 ) (2 ≤ i ≤ n) x i : state of neuron i n: number of neurons f(x) = tanh(x): output of neuron g: coupling gain (g > 0) τ: time constant

3. 3 Long transient oscillations were observed in the network of even neurons (n = 2m) with random initial condition. Simulation (n = 40, g = 10.0, τ= 1.0, x i (0) ~ N(0, 0.1 2 ) (1 ≤ j ≤ 40) ) Demonstration It was observed transient oscillations lasting more than a month in computer simulation with a workstation for n = 100. 4 5 2m 12 6 7 8 3 τdx 1 /dt = -x 1 + cf(x 2m ) τdx i /dt = -x i + cf(x i-1 ) (2 ≤ i ≤ 2m) x i : state of neuron i n: number of neurons f(x) = Tan -1 (x): output of neuron c: coupling constant (c < 0) τ: time constant x i (0) ~ i. i. d.

4. 4 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 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.

5. 5 2. Purpose of this study Observe the long transient oscillations in analog circuits of the ring networks of even neurons experimentally

6. 6 3. Circuit model Network of inverting amplifiers with diffusive coupling CRdV 1 /dt = -V 1 + V o (V n ) CRdV i /dt = -V i + V o (V i -1 ) (2 ≤ i ≤ n) V o (V in ) = -V p (V in > V p /g) -gV i (-V p /g ≤ V in ≤ V p /g) V p (V in < -V p /g) ±V p : power supply of OP amp g: gain of inverting amplifiers FIGURE 1. Analog circuit of a ring neuron model. Input-output relation of inverting amplifiers

7. 7 4. Experiment Number n of the amplifiers: 28, 32, 36, 40 V p = 12V, g = 10, C = 0.1μF, R = 10kΩ, (time constant: CR = 1ms) V i (t = 0): white noise with SD about 0.1V with an untuned AM radio FIGURE 2. Example of time series of the voltage V 1 in the analog circuit with 40 amplifiers.

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9. 9 Video1Video1 (n = 40, C = 0.1μF, R = 10kΩ, (time constant: CR = 1ms)) 10ms/div

10. 10 Video2Video2 (n = 40, C = 10μF, R = 10kΩ, (time constant: CR = 0.1s) ) Oscillation lasts about 45 minutes. 0.2s/div

11. 11 5. Properties of the duration of the transient oscillations 5.1 Exponential dependence of the duration on the number n of the neurons The duration of the transient oscillations increases exponentially as the number n of the amplifiers. T = 0.02x10 0.077n (sec)

12. 12 5.2 Effects of variations and fluctuations in the real circuits The duration of the transient oscillations obtained in the circuit experiment is shorter than that of computer simulation by a factor of 10. In simulation about 15% of transient oscillations lasted more than 100sec, which have not been observed in the experiment. This difference may be attributed to variations in the values of the elements or temporal fluctuations in voltage and currents since they destroy the symmetry of the system.

13. 13 Simulation with variations and fluctuations Time constant CR Gain g of the amplifiers Bias voltage of input to the amplifiers Temporal noise in voltage

14. 14 The main cause of decreases in the duration of the transient oscillations is variations in the biases of input to the amplifiers due to the offset voltages of OP amps. CRdV 1 /dt = -V 1 + V o (V n - V b(n) ) CRdV i /dt = -V i + V o (V i -1 - V b(i-1) ) (2 ≤ i ≤ n) V b(i) : random bias for each amplifiers The duration obtained with computer simulation adding the random biases V b(i) with the mean 0V and SD 0.1V matches the experimental results. However, these variations are rather large.

15. 15 6. Summary Long-lasting transient oscillations in ring networks of neurons with inhibitory one-way coupling were observed experimentally in the analog circuits with inverting amplifiers. Transient oscillations lasting about 25s with CR = 1ms and 45 minutes with CR = 0.1s were observed in the circuit of 40 amplifiers. The duration of the transient oscillations increases exponentially as the number of the amplifiers. The duration is expected to reach 10 6 seconds (more than 10 days) for 100 amplifiers with CR = 1ms. The duration of the transient oscillations observed in the circuits were shorter than that of simulation. This may be due to variations in bias voltages in the amplifiers but still remains unclear.