Controlling statistical moments of stochastic dynamical networks

We consider a general class of stochastic networks and ask which network nodes need to be controlled, and how, to stabilize and switch between desired metastable (target) states in terms of the first and second statistical moments of the system. We first show that it is sufficient to directly interfere with a subset of nodes which can be identified using information about the graph of the network only. Then we develop a suitable method for feedback control which acts on that subset of nodes and preserves the covariance structure of the desired target state. Finally, we demonstrate our theoretical results using a stochastic Hopfield network and a global brain model. Our results are applicable to a variety of (model) networks and further our understanding of the relationship between network structure and collective dynamics for the benefit of effective control.

Figure 1

Schematic of the control scheme. Left: State space of the stochastic dynamical network whose state is represented by the vector $\mathbf{x}$. Right: State space of the corresponding moments $\mathbf{\mu }$ (mean vector) and $\mathbf{C}$ (covariance matrix) of $\mathbf{x}$, whose dynamics is governed by a deterministic system of equations. Thick red arrows indicate pinning control through the subsets of nodes ${\mathbf{\mu }}_K, {\mathbf{C}}_K$, and ${\mathbf{x}}_K$, respectively. ${\mathbf{\mu }}_K^{*}$ and ${\mathbf{C}}_K^{*}$ are the values of dynamic variables ${\mathbf{\mu }}_K$ and ${\mathbf{C}}_K$ on the (metastable) target state. ${\mathbf{u}}_K(t,{\mathbf{x}}_J\left(t\right))$ denotes the feedback control signal which directly affects the subset of nodes ${\mathbf{x}}_K$ and depends on the complementary subset ${\mathbf{x}}_J$.

Figure 2

Subgraphs of connectivity graph ${\mathrm{\Gamma }}^{*}$ of the MS [cf. Eqs. (5a) and (5b)] of a network [cf. Eq. (1)] of $N=4$ nodes with circular graph $\mathrm{\Gamma }$. Left: Subgraph for the connectivity of the expected value nodes ${\mathbf{\mu }}_i$. Right: Subgraph for the connectivity between the (co)variance nodes ${\mathbf{C}}_{ij}, i,j\in \{1,\cdots ,N\}$. Numbers indicate the values of the indices of $\mathbf{\mu }$ and $\mathbf{C}$, respectively.

Figure 3

Schematic of mean vector $\mathbf{\mu }$ and covariance matrix $\mathbf{C}$, where a subset of nodes is highlighted by shaded background. That subset $K^{*}$ of nodes needs to be controlled, given that the FVS $K$ of the (original) network consists of nodes 2 and 3. In general, the set $K^{*}$ comprises the nodes ${\mathbf{\mu }}_i$ with the same index values as in the FVS of the original network, plus the corresponding rows and columns of the matrix $\mathbf{C}$; cf. Proposition t6.

Figure 4

Connectivity graph $\mathrm{\Gamma }$ of the example Hopfield network. Vertices that belong to the selected minimal FVS $\{1,4,7\}$ as well as their incoming and outgoing edges are highlighted in blue. There are no self-feedback loops because of the decay condition (t3). Note that $\mathrm{\Gamma }$ possesses a second minimal FVS of same cardinality: $\{1,2,4\}$.

Figure 5

Metastable states of the example Hopfield network, given by Eq. (19) (${\sigma }_{\mathrm{ext}}=0.01$). Values of mean vector $\mathbf{\mu }$ and correlation matrix $\tilde{\mathbf{C}}={\left[\text{diag}\left(\mathbf{C}\right)\right]}^{-\frac{1}{2}}\mathbf{C}{\left[\text{diag}\left(\mathbf{C}\right)\right]}^{-\frac{1}{2}}$, with covariance matrix $\mathbf{C}$, for the three states. Note that the three mean vectors are mutually distinct (approximately equidistant); the correlation matrices of state 1 and 3, however, are similar, but not the same.

Figure 6

Switching states ($1\to 2\to 3$) of the Hopfield network (${\sigma }_{\mathrm{ext}}=0.01$) by controlling the nodes of the FVS. Time tracks of (co)variances of (a) nodes $\{2,3,5\}$ and (b) the pinned nodes of the FVS $\{1,4,7\}$, obtained from Monte Carlo simulations (solid lines) and by solving the MS [Eqs. (17a) and (17b)] (dashed lines). Dotted lines indicate the (co)variances values for states 1 ($t\le 30$), 2 ($30<t\le 50$), and 3 ($t>50$). Arrows at $t=30$ and $t=50$ mark the initiation of control. The shaded region starting at $t=50$ (right) indicates the period of time during which the noise intensity of the control signal is set to zero temporarily $[{\mathbf{C}}_g\left(t\right)=0]$ to prevent violation of the Cauchy-Schwarz inequality (see Remark t7). Note that the Monte Carlo estimates of the pinned (co)variance values (i.e., those of ${\mathbf{u}}_K$) exhibit increased high (temporal) frequency because of the additive white Gaussian noise process of the control signal [cf. Eq. (10)].

Figure 7

Switching states ($1\to 2\to 3$) of the Hopfield network (${\sigma }_{\mathrm{ext}}=0.01$) by controlling the nodes $\{1,5,8\}$ (not a FVS). Time tracks of (co)variances of (a) nodes $\{3,6,7\}$ and (b) the pinned nodes $\{1,5,8\}$, obtained from Monte Carlo simulations (solid lines) and by solving the MS [Eqs. (17a) and (17b)] (dashed lines). Dotted lines indicate the (co)variances values for states 1 ($t\le 30$), 2 ($30<t\le 50$), and 3 ($t>50$). Arrows mark the initiation of control. Note that switching $1\to 2$ succeeds, but the controller fails to steer the system to state 3.

Figure 8

Switching states ($2\to 3$) of the Hopfield network (${\sigma }_{\mathrm{ext}}=0.01$) by controlling the nodes of the FVS $K=\{1,4,7\}$ using an open-loop controller [Eq. (20)]. Time tracks of (co)variances of nodes $\{2,3,5\}$, obtained from Monte Carlo simulations (solid lines) and by solving the MS [Eqs. (17a) and (17b)] (with ${\mathbf{\mu }}_K\left(t\right)\equiv {\mathbf{\mu }}_K^{*},{\mathbf{C}}_{KK}\left(t\right)\equiv {\mathbf{C}}_{KK}^{*},{\mathbf{C}}_{JK}\left(t\right)\equiv 0$, dashed lines). Dotted lines indicate the (co)variances values for states 2 ($t\le 20$) and 3 ($t>20$). The arrow marks the initiation of control. Note that the estimated (co)variance values (solid lines) follow the predictions from the MS solution (dashed lines) but do not converge to the target state.

Figure 9

Brain network model. (a) Visualization of the weighted 66-node network graph. Nodes of the selected minimal FVS are highlighted in blue. Thick ends of the edges denote edge destination and strength. (b) Maximum firing rate of the network as a function of the global coupling strength $G$. Colored markers denote two different target states. (c) Correlation matrices at the target states (upper triangles) and their mismatch from the actual result of switching control (lower triangles). Rows correspond to two choices of nodes for the feedback control method: minimal FVS and a random set. Columns correspond to the target states indicated in (a). Correlation matrices are obtained by temporally averaging results of Monte Carlo simulations ($2\times {10}^5$ iterations) over 1 s after the system has converged. (d) Normalized distance $\mathrm{\Delta }$ between the actual and the target states as a function of time, in terms of means and correlation matrices, for the feedback controller using FVS or random nodes, and for the clamp-release control scheme using FVS nodes. Note that diagonal elements are not considered when computing distance between correlation matrices (see Sec. 6). The system first evolves without control for 2 s. Then, the control method is applied to switch to the target high-activity state and 4 s later it is adjusted to switch to the resting state. For clamp-release control, release occurs 1.5 s after each initiation.