Almost-linear network

To illustrate the synaptic weights estimation in a neural spiking context, a simple feedfor-ward IF neuron network was simulated (depicted in Figure 3.2). Briefly, Neurons #2 − 5 were modeled by independent Poisson processes with firing rate λ. Neurons #8, 9 were modeled as single strong inputs by independent Poisson processes with firing rate 1.5λ.

Neuron #7 was implemented by a direct discrete time summation of the synaptic inputs α_i (mV), i = 1, . . . , 6 (i.e., the weighted outputs of Neurons #1−6 after some propagation delay), leading to its internal potential that was reset to V_reset = −80 (mV) and produced a spike when the threshold value V_th = −55 (mV) was reached. During the refractory period, the potential will linearly recover from V_reset to the resting potential V_rest = −70 (mV). Time resolution was set to be 1 ms and there was a 2 mV decrease/increase of the potential to the V_rest every unit time depending on the status of de/hyper -polarizations, respectively to model the diffusion of ions. The internal potential was forced to lie in the range [E_K⁺, E_{N a}⁺] = [−90, 60], the equilibrium potential of K⁺ and N a⁺, respec-tively and action potentials were normalized to 30 mV for display. Neurons #1, 6 were implemented in the same way that Neuron #7 was done with synaptic inputs 30 (mV) from Neurons #8, 9 respectively. Neurons #10, 11 were modeled as independent nodes by independent Poisson processes with firing rate λ.

Figure 3.2: A simple feedforward integrate-and-fire neuron network. Red circles repre-sent excitation and green squares reprerepre-sent inhibition. Neurons #2 − 5 are modeled by independent Poisson processes with firing rate λ. Neurons #8, 9 were modeled as single strong inputs by independent Poisson processes with firing rate 1.5λ. Neuron #7 is im-plemented by a direct discrete time summation of the synaptic inputs α₁, . . . , α₆(mV ).

Neurons #1, 6 are implemented in the same way that Neuron #7 is done with synaptic inputs 30 (mV) from Neurons #8, 9 respectively. Neurons #10, 11 are modeled as inde-pendent nodes by indeinde-pendent Poisson processes with firing rate λ. Neurons #5, 6 are inhibitory, i.e., α5, α6 < 0.

We begin with Simulation 1 in which the synaptic weights were fixed at α₁ = α₂ = α₃ = α₄ = 5 (mV) and α₅ = α₆ = −2.5 (mV), and the propagation delay of each source neuron was set to be 10 ms. 60 sec. voltage-trajectories of Neurons #1, 6, 7 were then simulated according to the way described above. The first 1 sec. of the trajectory of Neuron #7 with input rate λ = 40 Hz is shown in Figure 3.3 and the corresponding simulated spike train data is shown in Figure 3.4. The subthreshold trajectory of Neuron

#7 is not very regular due to the lack of self dynamics, compared to the nonlinear network (3.9a) introduced next. However, in this case it faithfully reflects the effects of the input neurons, that is, the actual degree of effects of the input neurons are to be proportional to the corresponding synaptic weights. To analyze the network directly through the simulated spike train data, a Gaussian kernel filtering with bandwidth 5 ms.

was performed to obtain an approximation of the subthreshold dynamics of each neuron in the network, the result is depicted in Figure 3.5. Based on the filtered data, both GC and NS indices were computed for different input rates λ =40, 60, and 80 Hz. In each case, the indices were both obtained from the average of 100 simulations and the results are summarized in Table 3.1.

We can find from Table 3.1 that although the GC indices correctly identify the di-rection of information flow among the network, the effects of excitations and inhibitions could not be differentiated directly by the sign of the indices since they are by definition

Figure 3.3: The first 1 sec. of a simulated voltage-trajectory of Neuron #7 with input rate λ = 40 Hz. The simulation was done according to the way described in the context with α1 = α2 = α3 = α4 = 5 (mV), α5 = α6 = −2.5 (mV), and 10 ms. propagation delay. The subthreshold trajectory is not very regular due to the lack of self dynamics, in other words, Neuron #7 is completely triggered by Neurons #1 − 6.

Figure 3.4: The first 1 sec. of a simulated spike train data of the simple feedforward network with λ = 40 Hz, α₁ = α₂ = α₃ = α₄ = 5 (mV), α₅ = α₆ = −2.5 (mV), and 10 ms. propagation delay.

to be nonnegative. From the GC indices, we can only tell that Neurons #2 − 4 have more influences than Neurons #1, 5 have on Neuron #7. Information about the underlying synaptic weights was not provided. As can be found in the lower part of Table 3.1, the synaptic weights were successfully reconstructed from the spike train data by the NS

Figure 3.5: A Gaussian kernel filtering with bandwidth 5 ms. was performed on the spike train data depicted in Figure 3.4 to obtain an approximation of the subthreshold dynamics of each neuron in the network. The computations of the GCI and NSI were based on the filtered results and this figure shows the first 250 data points.

Table 3.1: The numerical results of Simulation 1 in Section 3.3.2. The effects of excitations and inhibitions can be differentiated directly by the sign of the NS indices and the ratio of effects between them was close to 5.0 : −2.5 = 1 : −0.5. Numbers in parentheses are corresponding standard errors.

Input rate 1 → 7 2 → 7 3 → 7 4 → 7 5 → 7 6 → 7 8 → 1 9 → 6

Granger Causality Index (GCI)

λ = 40 0.0223 0.1972 0.1906 0.1919 0.0497 0.0066 2.9976 3.0085 (0.0035) (0.0125) (0.0155) (0.0129) (0.0059) (0.0031) (0.0868) (0.1016) λ = 60 0.0411 0.2770 0.2768 0.2769 0.0791 0.0115 2.5959 2.6063

(0.0038) (0.0138) (0.0141) (0.0156) (0.0076) (0.0020) (0.0573) (0.0829) λ = 80 0.0591 0.3466 0.3425 0.3444 0.1020 0.0156 2.3026 2.3235

(0.0080) (0.0188) (0.0155) (0.0168) (0.0083) (0.0037) (0.0531) (0.0575) Neuron Synaptic Index (NSI)

λ = 40 0.1155 0.1145 0.1140 0.1137 -0.0571 -0.0555 2.9972 3.0083 (0.0043) (0.0040) (0.0052) (0.0046) (0.0034) (0.0069) (0.0868) (0.1018) λ = 60 0.1530 0.1502 0.1505 0.1503 -0.0756 -0.0754 2.5959 2.6054

(0.0052) (0.0048) (0.0045) (0.0054) (0.0038) (0.0027) (0.0570) (0.0819) λ = 80 0.1799 0.1795 0.1793 0.1783 -0.0893 -0.0898 2.3030 2.3237

(0.0044) (0.0049) (0.0059) (0.0053) (0.0036) (0.0046) (0.0532) (0.0571)

indices in the sense that the ratio between excitatory and inhibitory sources was close to 5.0 : −2.5 = 1 : −0.5 for all different input rates. We note that the GC and NS indices from Neurons #8 − 12 to Neuron #7 are all zero (i.e., insignificant). As the results show, a large NSI does not necessarily imply a large GCI. That is, a strong synaptic trans-mission can not always guarantee a strong causal relationship; it depends also on the firing pattern/timing of the source and the coordination with other neurons. So, from

this perspective, NSI can be treated as a better proxy for synaptic weights rather than a new causality measure. GCI provides information on causal structure while NSI provides complementary information on synaptic transmission.

In Simulation 2, the input rate λ was fixed at 60 Hz while the synaptic weights varied.

Let α₁ = α₂ = α₃ = α₄ = 5 (mV) and α₅ = α₆ = −k × 5 (mV). Three different weight-ratio k = 0.5, 1.0, 1.5 were considered, and the computed NS indices are presented in Table 3.2. We can find that the ratio between excitatory and inhibitory sources was still close to 1 : −k as weight-ratio changes.

Table 3.2: The numerical results of Simulation 2 in Section 3.3.2. The input rate λ was fixed at 60 Hz, α₁ = α₂ = α₃ = α₄ = 5 (mV) , and α₅ = α₆ = −k × 5 (mV). The ratio of effects between excitatory and inhibitory sources was still close to 1 : −k as weight ratio changes. Numbers in parentheses are corresponding standard errors.

weight ratio 1 → 7 2 → 7 3 → 7 4 → 7 5 → 7 6 → 7 8 → 1 9 → 6 Neuron Synaptic Index (NSI)

k = 0.5 0.1530 0.1502 0.1505 0.1503 -0.0756 -0.0754 2.5959 2.6054 (0.0052) (0.0048) (0.0045) (0.0054) (0.0038) (0.0027) (0.0570) (0.0819) k = 1.0 0.1077 0.1072 0.1063 0.1067 -0.1025 -0.1031 2.5928 2.6151

(0.0041) (0.0031) (0.0044) (0.0043) (0.0034) (0.0037) (0.0857) (0.0671) k = 1.5 0.0615 0.0621 0.0612 0.0625 -0.0857 -0.0866 2.5849 2.5869

(0.0030) (0.0031) (0.0036) (0.0037) (0.0029) (0.0039) (0.0764) (0.0735)