Inferential framework for nonstationary dynamics. II. Application to a model of physiological signaling

The problem of how to reconstruct the parameters of a stochastic nonlinear dynamical system when they are time-varying is considered in the context of online decoding of physiological information from neuron signaling activity. To model the spiking of neurons, a set of FitzHugh-Nagumo (FHN) oscillators is used. It is assumed that only a fast dynamical variable can be detected for each neuron, and that the monitored signals are mixed by an unknown measurement matrix. The Bayesian framework introduced in paper I (immediately preceding this paper) is applied both for reconstruction of the model parameters and elements of the measurement matrix, and for inference of the time-varying parameters in the nonstationary system. It is shown that the proposed approach is able to reconstruct unmeasured (hidden) slow variables of the FHN oscillators, to learn to model each individual neuron, and to track continuous, random, and stepwise variations of the control parameter for each neuron in real time.

Figure 1

Numerical simulation of the FitzHugh-Nagumo oscillator (1). (a) Examples of the time traces of $v_j$ (solid line) and $q_j$ (dashed line). (b) Nullclines are shown by the dashed (first equation) and dotted (second equation) lines, and the corresponding phase trajectory is shown by the thin solid line.

Figure 2

(Color online) Time-series data generated by the model (1),(2) (a) before and (b) after mixing, for the parameters given in Table . Parameters ${\eta }_1$ and ${\eta }_2$ fluctuate between 0.35 and 0.45. The dotted (red) lines show $v_1\left(t\right)$ and $y_1\left(t\right)$ and the solid (blue) lines show $v_2\left(t\right)$ and $y_2\left(t\right)$.

Figure 3

Typical example of the convergence of parameters as a function of signal length. ${\tilde{\eta }}_1$ and ${\tilde{b}}_{222}$ are plotted as functions of time, i.e., of the number of data. The first point corresponds to a block of 5000 data points; each successive point corresponds to an additional 5000 data, as discussed in the text. Vertical bars show standard deviations of the inferred values, calculated over 1000 realizations. The horizontal dashed lines indicate the true values of the model parameters as given in Tables .

Figure 4

(a) and (b) Typical examples of convergence for two components of the measurement matrix $\mathit{X}$ as a function of the measurement time $t$. The model parameters for this numerical test are given in Table . We have used nine data blocks with 5000 points in each block. The standard deviations of the inferred parameters shown by the vertical bars are calculated over 1000 realizations. The horizontal lines show the true values of the model parameters. The sampling rate was $35\mathrm{kHz}$.

Figure 5

(a) and (b) Convergence of the control parameters ${\eta }_1$ and ${\eta }_2$ as functions of the measurement time $t$. Values of the model parameters for this numerical test are given in Table . We have used nine data blocks with 5000 points in each block. The standard deviations of the inferred parameters shown by the vertical bars are calculated over 1000 realizations. The horizontal lines show the true values of the control parameters. The sampling rate was $35\mathrm{kHz}$. The insets in both figures show the decay of the largest eigenvalue of ${\widehat{\mathbf{\Xi }}}^{-1}$.

Figure 6

(Color online) The largest eigenvalues ${\lambda }_i$ of the matrix ${\widehat{\mathbf{\Xi }}}^{-1}$ under different assumptions: (i) when none of the coefficients of the dynamics in Eq. (4) are known (full red lines); (ii) when the coefficients of the third and second powers are known (dashed blue lines); (iii) when all parameters except ${\eta }_i$ are known (dotted black lines). The dynamical coefficients are the same as in Fig. 4. The number of runs to obtain the averaged convergence was 1000 for each data block size. The actual distribution for each eigenvalue is highly asymmetric over the number of the runs, and typical values of ${\lambda }_i$ are lower than their respective means.

Figure 7

Typical example of the fast convergence of the control parameters (a) ${\eta }_1$ and (b) ${\eta }_2$, as functions of time (length of signal). The first point corresponds to 200 data points in one block. For each next point, the number of data points was increased by 200. The vertical lines show the standard deviations of the inferred values of the control parameters calculated over 1000 runs. The horizontal dashed lines indicate the true values of the parameters. The mixing matrix is defined in Table . The inferred parameters ${\eta }_i$ start from an initial value of ${\eta }_1={\eta }_2=0.2$ and converge quickly to the true values of ${\eta }_1=0.4$, ${\eta }_2=0.3$. The coefficients ${\alpha }_i$, ${\beta }_i$, ${\gamma }_i$, and $d_{ij}$ are given in Table . The noise amplitude is $\sqrt{d_1}=\sqrt{d_2}=0.01225$.

Figure 8

(Color online) Inference of the parameters of two uncoupled FHN systems mixed by the measurement matrix. It is assumed that ${\eta }_1$ and ${\eta }_2$ change stepwise while all other parameters of the system are fixed and unknown. (a) The inferred values of ${\eta }_1$ (dashed red lines) are compared with their true values (full blue lines). (b) Measured mixed values of the coordinate $x_1\left(t\right)$. (c) Inferred values of the coordinate $q_1\left(t\right)$ (red dotted line) are compared with its true values (blue solid line). The other parameters are fixed at the values given in Table . The noise amplitude is $\sqrt{d_1}=\sqrt{d_2}=0.01225$.

Figure 9

(Color online) Inference of the model parameters of two uncoupled FHN systems mixed by the measurement matrix. It is assumed that ${\eta }_1$ and ${\eta }_2$ change stepwise while all other parameters of the system are fixed and known. (a) Inferred values ${\eta }_1$ (short elements of red dashed line) are compared with their true values (short elements of full blue line) as a function of time. (b) The time trace of the measured coordinate $x_1\left(t\right)$. (c) The time trace of the inferred coordinate ${\tilde{q}}_1\left(t\right)$ (red dotted line) is compared with its true value $q_1\left(t\right)$ (blue solid line). The values of the other parameters are fixed, as given in Table . The noise amplitude is $\sqrt{d_1}=\sqrt{d_2}=0.01225$.

Figure 10

(Color online) Inference of ${\eta }_1$ and ${\eta }_2$ while smoothly varying in the presence of noise. No prior knowledge of the model parameters is assumed. (a) The inferred values of ${\eta }_1$ (dashed red lines) are compared with their true values (full blue lines). (b) The measured time trace of the mixed coordinate $x_1\left(t\right)$. (c) The inferred time trace of the mixed coordinate ${\tilde{q}}_1\left(t\right)$ (dashed red line) is compared with its true value $q_1\left(t\right)$ (full blue line). The values of the other parameters are given in Table . The noise amplitude is $\sqrt{d_1}=\sqrt{d_2}=0.01225$.