Learning spiking neuronal networks with artificial neural networks: neural oscillations

Abstract

First-principles-based modelings have been extremely successful in providing crucial insights and predictions for complex biological functions and phenomena. However, they can be hard to build and expensive to simulate for complex living systems. On the other hand, modern data-driven methods thrive at modeling many types of high-dimensional and noisy data. Still, the training and interpretation of these data-driven models remain challenging. Here, we combine the two types of methods to model stochastic neuronal network oscillations. Specifically, we develop a class of artificial neural networks to provide faithful surrogates to the high-dimensional, nonlinear oscillatory dynamics produced by a spiking neuronal network model. Furthermore, when the training data set is enlarged within a range of parameter choices, the artificial neural networks become generalizable to these parameters, covering cases in distinctly different dynamical regimes. In all, our work opens a new avenue for modeling complex neuronal network dynamics with artificial neural networks.

Appendix

1.1 Tau-leaping and SSA algorithms

The simulations of the SNN dynamics are carried out by two algorithms: Tau-leaping and Stochastic Simulation Algorithm (SSA). The key difference is that, The tau-leaping method processes events that happen during a time step $\tau$ in bulk, while SSA simulates the evolution event by event. Of the two, tau-leaping can be faster (with properly chosen $\tau$), while SSA is usually more precise with the precision that scales with C++ execution. Here we illustrate a Markov jump process as an example.

Algorithms. Consider $X(t)=\{x_1(t),x_2(t),..., x_N(t)\}$, where X(t) can take values in a discrete state space

$$\begin{aligned} S=\{s_1,s_2,\ldots ,s_M\subset \mathbb {R}^N\}. \end{aligned}$$

The transition from state X to state $s_i$ at time t is denoted as $T_{s_i}^t(X)$, taking an exponential distributed waiting time with rate $\lambda _{s_i\leftarrow X}$. Here, $s_i\in S(X)$ which are states adjacent to state X with a non-zero transition probability. For simplicity, we assume $\lambda _{s_i\leftarrow X}$ does not explicitly depend on t except via X(t).

Tau-leaping only considers X(t) on a time grid $t = jh$, for $j = 0,1,...,T/h$, assuming state transfer occurs for at most one time within each step:

$$\begin{aligned}&P(X^{(j+1)h} = s_i) \\&\qquad ={\left\{ \begin{array}{ll} h\lambda _{s_i\leftarrow X^{jh}} &{} \forall s_i\in S(X^{jh}), \\ 1-h\sum _{s_i\in S(X^{jh})}\lambda _{s_i\leftarrow X^{jh}} &{} s_i = X^{jh}, \\ 0 &{} \text {otherwise}, \end{array}\right. } \end{aligned}$$

On the other hand, SSA accounts for this simulation problem as:

$$\begin{aligned} X(T) = T_{X_k}^{t_k}\circ T_{X_{k-1}}^{t_{k-1}}\circ \ldots \circ T_{X_1}^{t_1}(X(0)), \end{aligned}$$

i.e., starting from $X^0$, X transitions to $X_1, X_2,\ldots , X_k = X(T)$ at time $0<t_1< t_2<\ldots< t_k <T$.

For $t_\ell<t<t_{\ell +1}$, we sample the transition time from $\text {Exp}(\sum _{s_i\in S(X(t))} \lambda _{s_i\leftarrow X(t)})$. That is, for independent, exponentially distributed random variables

$$\begin{aligned} \tau _i\sim \text {Exp}(\lambda _{s_i\leftarrow X(t)}), \end{aligned}$$

we have

$$\begin{aligned} t_{\ell +1} - t_\ell = \min _{s_i\in S(X(t))}\tau _i\sim \text {Exp}\left( \sum _{s_i\in S(X(t))} \lambda _{s_i\leftarrow X(t)}\right) . \end{aligned}$$

Therefore, in each step of an SSA simulation, the system state evolves forward by an exponentially distributed random time, whose rate is the sum of rates of all exponential “clocks". Then we randomly choose the exact state $s_i$ to which transition takes place with probability weighted by the sizes of the pending events.

Implementation on spiking networks. We note that X(t) will changes when

• 1. neuron i receives external input ($v_i$ goes up for 1, including entering $\mathcal {R}$);

• 2. neuron i receives a spike ($H^E_i$ or $H^I_i$ goes up for 1);

• 3. a pending spike takes effect to neuron i ($v_i$ goes up/down according to synaptic strengths);

• 4. neuron i walks out from refractory ($v_i$ goes from $\mathcal {R}$ to 0).

The corresponding transition rates are directly given ($\lambda ^E$ and $\lambda ^I$) or the inverses of the physiological time scales ($\tau ^{E}$, $\tau ^{I}$, and $\tau ^{\mathcal {R}}$). In an SSA simulation, when the state transition elicits a spike in a neuron, the synaptic outputs generated by this spike are immediately added to the pool of corresponding types of effects, and the neuron goes into the refractory state. However, in a tau-leaping simulation, the spikes are recorded but the synaptic outputs are processed in bulk at the end of each time step. Therefore, all events within the same time step are uncorrelated.

1.2 The coarse-graining mapping

Here we give the definition of the coarse-grained mapping $\mathcal {C}$ in Eq. 6. For $\forall \omega \in \varvec{\Omega }$ that

$$\begin{aligned} \omega =&(V_1,\ldots ,V_{N_E},V_{N_E+1},\ldots ,V_{N_E+N_I},\\&H^E_1,\ldots ,H^E_{N_E},H^E_{N_E+1},\ldots ,H^E_{N_E+N_I},\\&H^I_1,\ldots ,H^I_{N_E},H^I_{N_E+1},\ldots ,H^I_{N_E+N_I}), \end{aligned}$$

we define

$$\begin{aligned} \mathcal {C}(\omega )=\tilde{\omega }=(&n^E_1, n^E_2, \ldots , n^E_{22}, n^E_R, n^I_1, n^I_2, \ldots , n^I_{22}, n^I_R,\\&H^{EE}, H^{EI}, H^{IE}, H^{II}), \end{aligned}$$

where,

$$\begin{aligned} n^E_i&=\sum _{j=1}^{N_E}{} {\textbf {1}}_{\varGamma _i}(V_j),\quad \text {for }i=1,\ldots ,22;\\ n^E_R&=\sum _{j=1}^{N_E}{} {\textbf {1}}_{\{\mathcal {R}\}}(V_j);\\ n^I_i&=\sum _{j=N_E+1}^{N_E+N_I}{} {\textbf {1}}_{\varGamma _i}(V_j),\quad \text {for }i=1,\ldots ,22;\\ n^I_R&=\sum _{j=N_E+1}^{N_E+N_I}{} {\textbf {1}}_{\{\mathcal {R}\}}(V_j); \end{aligned}$$

and

$$\begin{aligned} H^{EE}&=\sum _{j=1}^{N_E}H_j^E;\qquad H^{IE}=\sum _{j=N_E+1}^{N_E+N_I}H_j^E;\\ H^{EI}&=\sum _{j=1}^{N_E}H_j^I;\qquad H^{II}=\sum _{j=N_E+1}^{N_E+N_I}H_j^I. \end{aligned}$$

Here, ${\textbf {1}}_{\text {A}}(a)$ is an indicator function of set A, i.e., ${\textbf {1}}_{\text {A}}(a) = 1$ $\forall a\in \text {A}$, otherwise ${\textbf {1}}_{\text {A}}(a) = 0$. $\varGamma _i$ is a subset of the state space for membrane potential, and

$$\begin{aligned} \varGamma _i = [-15+5i, -10+5i)\cap \varGamma . \end{aligned}$$

1.3 Pre-processing surrogate data: discrete cosine transform

Here we explain how discrete cosine transform (DCT) works in the pre-processing. For an input probability mass vector

$$\begin{aligned} \varvec{p} = (n_1, n_2, \cdots , n_{22}), \end{aligned}$$

its DCT output $\mathcal {F}_c(\varvec{p}) = (c_1,c_2,..., c_{22})$ is given by

$$\begin{aligned} c_k = \sqrt{\frac{2}{22}}\sum _{l=1}^{22} \frac{n_l}{\sqrt{1+\delta _{kl}}}\cos \left( \frac{\pi }{2N}(2l-1)(k-1)\right) , \end{aligned}$$

(9)

where $\delta _{kl}$ is the Kronecker delta function. The iDCT mapping $\mathcal {F}^{-1}_c$ is defined as the inverse function of $\mathcal {F}_c$.

1.4 The linear formula for firing rates

When preparing the parameter-generic training set, we use simple, linear formulas to estimate the firing rate of E neurons and I neurons ($f_E$ and $f_I$, see Li and Hui 2019; Li et al. 2019). We take $\theta \in {\Theta }$ for the synaptic coupling strength, while other constants are the same as in Table 1.

$$\begin{aligned} f_E=\frac{\lambda ^E(M+C^{II})-\lambda ^I C^{EI}}{(M-C^{EE})(M+C^{II})+(C^{EI}C^{IE})}\\ {f_I=\frac{\lambda ^I(M-C^{EE})+\lambda ^E C^{IE}}{(M-C^{EE})(M+C^{II})+(C^{EI}C^{IE})}} \end{aligned}$$

where

$$\begin{aligned}&C^{EE}=N^EP^{EE}S^{EE},\ C^{IE}=N^EP^{IE}S^{IE}\\&C^{EI}=N^IP^{EI}S^{EI},\ C^{II}=N^IP^{II}S^{II}. \end{aligned}$$

1.5 The deep network architecture

In general, artificial neural networks (ANNs) are interconnected computation units. Many different architectures are possible for ANNs; in this paper, we adopt the feedforward deep network architecture, which is one of the simplest (Fig. 7).

Fig. 7

Diagram of a feedforward ANN

A feedforward ANN has a layered structure, where units in the $i-$th layer drive the $(i+1)-$th layer with a weight matrix $\varvec{W}_i$ and a bias vector $b_i$. Computation is processed from one layer to the next. The first, “input layer" takes an input vector x, sending its output $\varvec{W}_1x+b_1$ to the first "hidden layer"; the first hidden layer then sends output $\varvec{W}_2 f(\varvec{W}_1x+b_1)+b_2$ to the next layer, and so on, until the last, “output layer" produces an output vector y. In this paper, we implemented a feedforward ANN with four layers containing 512, 512, 512, and 128 neurons, respectively. We chose the Leaky ReLU function with a default negative slope of 0.01 as our activation function $f(\cdot )$.

The training of feedforward ANNs is achieved by the back-propagation (BP) algorithm. Let $\mathcal{N}\mathcal{N}(x)$ denote the prediction of the ANN with input x, and $L(\cdot )$ the loss function. With each entry (x, y) in the training data, we minimize the loss $L(y-\mathcal{N}\mathcal{N}(x))$ following the gradients on each dimension of $W_i$ and $b_i$. The computation of gradients takes place from the last layer $W_n$’s and $b_n$, then “propagated back" to adjust previous $W_i$ and $b_i$ on each layer. We chose the mean-square error as our loss function, i.e. $L(\cdot )=||\cdot ||_{L^2}^2.$

Fig. 8

Left: pre-, post- and predicted MFE profiles without DCT + iDCT; Middle: pre-, post- and predicted MFE profiles with DCT + iDCT; Right: pre-, post- and predicted MFE profiles with network parameters as additional inputs of ANN. (Left and Middle: $S^{EE}$, $S^{IE}$, $S^{EI}$, $S^{II}$ = 4, 3, $-$2.2, -2; Right: 3.82, 3.24, $-$2.05, $-$1.87.)

1.6 Pre-processing in ANN predictions

Here we provide more examples of the ANN predictions of the voltage profiles. We compare how ANNs predict post-MFE voltage distributions $\varvec{p}^E$ and $\varvec{p}^I$ in three different settings in Fig. 8. In each panel divided by red lines, the left column gives an example of pre-MFE voltage distributions, while the right column compares the corresponding post-MFE voltage distributions collected from ANN predictions (red) vs. SSA simulation. Results from ANNs without pre-processing, with pre-processing, and the parameter-generic ANN are depicted in the left, middle, and right panels.

1.7 Principal components of voltage distributions

The voltage distribution vectors in the form below are used to plot the distribution in the phase space as shown in middle panels of Figs. 5, 6, 9D, 10D, 11D, 12D, 13, 14, and 15

$$\begin{aligned} (&n^E_1, n^E_2, \ldots , n^E_{22}, n^E_R, n^I_1, n^I_2, \ldots , n^I_{22}, n^I_R) \end{aligned}$$

The vectors from the training set (colored in blue in figures) are selected to generate the basis of the phase space through svd function in numpy.linalg in Python. The first two rows of the $V^\top$ are the first two PCs of the space. The scores of vectors from the training set and approximated results are dot products of these vectors and the normalized PCs.

The plain ks-density function in MATLAB is used to estimate the kernel smoothing density of the profile distribution based on the data points generated above. The contours show the level of each tenth of the maximal height (with 0.1% bias for demonstrating the top) in the distributions.

Fig. 9

DNN predictions and surrogate dynamics in ER random network with 400 neurons. A. Mapping $\widehat{F}^{\theta }_1$ in ER network for $\theta = (S^{EE}, S^{EI}, S^{IE}, S^{II})= (4, 3, -2.2, -2)$. A. Left: a pre-MFE $\varvec{p}^E(v)$ and a $\varvec{p}^I(v)$; Right: post-MFE $\varvec{p}^E(v)$ and $\varvec{p}^I(v)$ produced by ANN (blue) vs. spiking network simulations (orange). B. Comparison of E and I spike number during MFEs, ANN predictions vs. SSA simulations. The distributions are depicted by 10th contours of max in ks-density estimation; C. Example of pre and post-MFE voltage distributions $\varvec{p}^E$ and $\varvec{p}^I$ in the surrogate dynamics. D. Distribution depicted by 10th-contours of the first two principal components of $\varvec{p}^E$ and $\varvec{p}^I$. E. Raster plots of simulated surrogate dynamics and the real dynamics starting from the same initial profiles (color figure online)

Fig. 10

DNN predictions and surrogate dynamics in a 400-neuron random network with log-normal degree distribution. A-E are in parallel to Fig. 9

Fig. 11

DNN predictions and surrogate dynamics in ER random network with 4000 neurons. A-E are in parallel to Fig. 9, except that $\theta = (S^{EE}, S^{EI}, S^{IE}, S^{II})= (0.4, 0.3, -0.22, -0.2)$

Fig. 12

DNN predictions and surrogate dynamics in a 4000-neuron random network with log-normal degree distribution. A-E are in parallel to Fig. 11

Fig. 13

Surrogate dynamics produced by parameter-generic MFE mapping $\widehat{F}_1$ in two fixed networks with 400 neurons. Left: ER network; Right: Network with log-normal degree distribution.A-B: Example of pre and post-MFE voltage distributions $\varvec{p}^E$ and $\varvec{p}^I$ in the surrogate dynamics. C-D: Distribution depicted by 10th-contours of the first two principal components of $\varvec{p}^E$ and $\varvec{p}^I$. E-F: Raster plots of simulated surrogate dynamics and the real dynamics starting from the same initial profiles

Fig. 14

Surrogate dynamics produced by parameter-generic MFE mapping $\widehat{F}_1$ in two fixed networks with 4000 neurons. Left: ER network; Right: Network with log-normal degree distribution. A-F are in parallel to Fig. 13

Fig. 15

Surrogate dynamics with a parameter set that lies out of the sampling 4D cube produced by parameter-generic MFE mapping $\widehat{F}_1$. Left: $(S^{EE},S^{IE},S^{EI},S^{II}) = (5, 3, -2.2, -2)$; Right: $(S^{EE},S^{IE},S^{EI},S^{II}) = (4, 4, -2.2, -2)$. A-F are in parallel to Fig. 13

Fig. 16

Fixed random graphs used for SNN simulation. ER: Erd?s-Rényi random graph. LN: random graphs with log-normal degree distribution. A light-colored block at (i, j) represents a directed synaptic connection from i to j. The last quarter of the neurons are inhibitory, while the others are excitatory. This figure is compressed to reduce the file size

Fig. 17

Distributions of the magnitude of MFEs (number of spikes) from sampled parameters sets and the specific parameter set $(S^{EE},S^{IE},S^{EI},S^{II}) = (4, 3, -2.2, -2)$

Fig. 18

Distributions of the magnitude of MFEs (number of spikes) from the enlarged set of initial profiles and a single trajectory

Fig. 19

Firing rates of networks with parameter sets that can produce accepted MFEs for training. (n=3000)

1.8 Consistent results from fixed random networks

Here we test the capability of our method with fixed network architectures. We select two types of random graphs: 1. Erd?s-Rényi random graph (ER), and 2. random graphs with log-normal degree distribution (LN). In both types of graphs, the average edge density is consistent with Ps in Table 1. Their adjacency matrices are shown in Fig. 16. The sampling of four types of edges in LN random graphs leverages a standard deviation of 0.2 in logarithm and constraints from the mean degrees. Results are obtained from both sizes, 400 and 4000, of the random graphs. For each network, MFEs are captured from original network simulations and simulations with enlarged initial profiles (Fig. 18). The trained parameter-specific MFE mappings $\widehat{F}^{\theta }_1$ are used to produce predictions of post-MFE states and surrogate network dynamics (Figs. 9, 10, 11, 12). MFEs are further captured in simulations with various parameter sets sampled from the 4D cube $\varvec{\Theta }$. The trained parameter-generic $\widehat{F}_1$ is used to produce predictions and surrogate dynamics, which are shown in Figs. 13 and 14. As seen in Figs. 9, 10, 11, 12, 13 and 14, the performance of the ANN surrogate is consistent with the case when postsynaptic connections are decided on-the-fly.

1.9 Varying the synaptic coupling strengths generates a broad range of firing rates and magnitudes of MFEs

The training of parameter-generic MFE mapping $\widehat{F}_1$ needs MFEs from simulations with a variety of sets of parameter $\theta$. As introduced in Sect. 4.3, the sets sampled from the 4D cube $\varvec{\Theta }$ are first filtered by the estimated firing rate computed by the linear formula. A large fraction (about 80%) of the sets from the previous step generate MFEs that can be captured and accepted by our algorithm. These sets generate a wide range of firing rates (Fig. 19). The simulated firing rates match the linear formula in general. The major rejected region appears with high $S^{EE}$, $S^{IE}$ and low $S^{II}$, $S^{EI}$, which is in the neighborhood with the accepted region with high firing rate and near the singular region of the linear formula.

The magnitudes of MFEs (number of spikes) from various parameters show a much wider distribution than the original set of parameters (Fig. 17).

1.10 Extrapolating network dynamics out of $\varvec{\Theta }$ with parameter-generic MFE mapping $\widehat{F}_1$

To test the extrapolation ability of our method, we generate surrogate dynamics in networks with $\theta$’s outside of $\varvec{\Theta }$ with the parameter-generic MFE mapping $\widehat{F}_1$. The two $\theta$’s are $(S^{EE},S^{IE},S^{EI},S^{II}) = (5, 3, -2.2, -2)$ and $(S^{EE},S^{IE},S^{EI},S^{II}) = (4, 4, -2.2, -2)$.

Parameter-generic MFE mapping $\widehat{F}_1$ trained with MFEs in $\varvec{\Theta }$ can still capture the neuronal oscillations and predict post-MFE states in the two networks. (Fig. 15) Behaviors of networks under these two sets of parameters are also reproduced. Strong recurrent excitation in the former $\theta$ makes MFEs readily to be concatenated, while the less-synchronized character of the latter $\theta$ makes MFEs hard to trigger or identify. This result shows the robustness and capability of extrapolation of our method, while our future work can focus on improving the precision of prediction in detail (Figs. 18, 19).