13.2.1 Theory

Due to the short timescales of STSP, it can be studied as a form of nonlinear dynamics. In this context, dynamics refers to finite-memory history dependency of PSPs, i.e., the magnitude of PSP is influenced not only by the current input spike or stimulation, but also preceding inputs within a memory window. Nonlinear dynamics happens when the dynamical interaction does not obey the additivity property. Such a system can be expressed by a time-invariant function (Fig. 13.2, top panel).

We use a nonlinear dynamical and biologically plausible model structure to extract PSP and STSP from input–output spiking activities [9,56–59,62,65]. This is a nontrivial task since both inputs and output are point-process signals containing only spike timings; PSP and STSP need to be inferred instead of directly measured. The model takes the form of a multiinput, single-output (MISO) model that can be expressed as (Fig. 13.3)

$w=u(K,x)+a(H,y)+\epsilon (\sigma ),$

$y=\{\begin{array}{cc}\hfill 0\hfill & \hfill \text{when }w<\theta ,\hfill \\ \hfill 1\hfill & \hfill \text{when }w\ge \theta .\hfill \end{array}$

Variable x represents input spike trains; y represents the output spike train; w represents the prethreshold membrane potential of the output neurons, which is expressed as the summation of the postsynaptic potential u caused by input spike trains, the output spike–triggered after-potential a and a Gaussian white noise ε with standard deviation σ. A threshold, θ, determines the generation of the output spike and the associated feedback after-potential (a).

The feedforward transformation from x to u and the feedback transformation from y to a take the form of a second-order Volterra model K and a first-order Volterra model H, respectively. We have

$\begin{array}{c}u(t)=k_0+\sum _{n=1}^N\sum _{\tau =0}^{M_k}{k}_1^{(n)}(\tau )x_n(t-\tau )+\sum _{n=1}^N\sum _{{\tau }_1=0}^{M_k}\sum _{{\tau }_2=0}^{M_k}{k}_{2s}^{(n)}({\tau }_1,{\tau }_2)x_n(t-{\tau }_1)x_n(t-{\tau }_2)\hfill \\ \phantom{u(t)}+\sum _{n_1=1}^N\sum _{n_2=1}^{n_1-1}\sum _{{\tau }_1=0}^{M_k}\sum _{{\tau }_2=0}^{M_k}{k}_{2x}^{(n_1,n_2)}({\tau }_1,{\tau }_2)x_{n_1}(t-{\tau }_1)x_{n_2}(t-{\tau }_2)+...,\hfill \end{array}$

$a(t)=\sum _{\tau =1}^{M_h}h(\tau )y(t-\tau ).$

The zero order kernel, $k_0$, is the value of u when there is no input. It determines the spontaneous firing rate of the output neuron. First-order kernels $k_1^{(n)}$ describe the linear relation between the nth input $x_n$ and u, as functions of the time intervals (τ) between the present time and the past time. Second-order self-kernels $k_{2s}^{(n)}$ describe the nonlinear interactions between pairs of spikes within the nth input $x_n$ as they affect u. Second-order cross kernels $k_{2x}^{(n_1,n_2)}$ describe the nonlinear interactions between pairs of spikes from different inputs ($x_{n_1}$ and $x_{n_2}$) as they affect u. N is the number of inputs. Higher-order kernels can also be included to capture higher-order nonlinearities. The term h represents a linear feedback kernel that transforms the output spike to the after-potential a. $M_k$ and $M_h$ are memory lengths of feedforward and feedback processes, respectively.

To reduce number of parameters and avoid overfitting, basis functions are utilized in model estimations [62]. Basis functions b can take the forms of Laguerre basis [42,43,60] or B-spine basis [22]. With input and output spike trains x and y convolved with b, we have

$v_j^{(n)}(t)=\sum _{\tau =0}^{M_k}{b}_j(\tau )x_n(t-\tau ),$

$v_j^{(h)}(t)=\sum _{\tau =1}^{M_h}{b}_j(\tau )y(t-\tau ).$

Eqs. (13.3) and (13.4) are rewritten

$\begin{array}{c}u(t)=c_0+\sum _{n=1}^N\sum _{j=1}^L{c}_1^{(n)}(j)v_j^{(n)}(t)+\sum _{n=1}^N\sum _{j_1=1}^L\sum _{j_2=1}^{j_1}{c}_{2s}^{(n)}(j_1,j_2)v_{j_1}^{(n)}(t)v_{j_2}^{(n)}(t)\hfill \\ \phantom{u(t)}+\sum _{n_1=1}^N\sum _{n_2=1}^{n_1-1}\sum _{j_1=1}^L\sum _{j_2=1}^L{c}_{2x}^{(n_1,n_2)}(j_1,j_2)v_{j_1}^{(n_1)}(t)v_{j_2}^{(n_2)}(t)+...,\hfill \end{array}$

$a(t)=\sum _{j=1}^L{c}_h(j)v_j^{(h)}(t),$

where $c_1^{(n)}$, $c_{2s}^{(n)}$, $c_{2x}^{(n_1,n_2)}$ and $c_h$ are the model coefficients of $k_1^{(n)}$, $k_{2s}^{(n)}$, $k_{2x}^{(n_1,n_2)}$ and h, respectively, and $c_0$ is equal to $k_0$. Given the kernels are smooth and continuous functions, the number of basis functions L is much smaller than the memory length ($M_k$ and $M_h$).

In Eqs. (13.7) and (13.8), v and vv can be calculated from the known x, y and b. The Volterra series essentially expresses the nonlinear relationship between u and x into a linear relationship between u and [v, vv]. The joint effect of the threshold θ and the Gaussian noise ε is equivalent to a probit link function that maps the value of $u+a$ into the probability of y is equal to 1. The whole model thus can be expressed as a generalized linear model with the nonlinearity structured in the Volterra series. Therefore, this MISO model can be termed a generalized Volterra model (GVM) [58,59]. As a special case, the first-order GVM is equivalent to the commonly used generalized linear models [50,49,52,69,26,19,73].

Due to the point-process nature of the input and output signals, all model variables are dimensionless. In addition, the values of k, h, u, a, w, ε, σ and θ can be scaled; θ and $k_0$ can be translated, without influencing the probability of generating output spikes. So, in practice, $k_0$, $k_1$, $k_{2s}$, $k_{2x}$ and h are first estimated with a unitary σ and a zero-valued θ using the Matlab^® glmfit function and then normalized with the absolute value of $k_0$. In the final format, σ is equal to $1/|k_0|$; θ is equal to zero; the baseline value of w (i.e., $k_0$) is −1 [57,58]. The Laguerre parameter controlling the asymptotic decaying rate of the basis functions and the total number of basis functions L are optimized with respect to the log-likelihood function [62].

The Volterra kernels quantitatively describe the input–output nonlinear dynamics of the neuron. A more intuitive representation is given by the single-pulse and paired-pulse response functions ($r_1$ and $r_2$) derived from the kernels. We have

$r_1^{(n)}(\tau )=k_1^{(n)}(\tau )+k_{2s}^{(n)}(\tau ,\tau ),$

$r_{2s}^{(n)}({\tau }_1,{\tau }_2)=2k_{2s}^{(n)}({\tau }_1,{\tau }_2),$

$r_{2x}^{(n_1,n_2)}({\tau }_1,{\tau }_2)=k_{2x}^{(n_1,n_2)}({\tau }_1,{\tau }_2),$

where $r_1^{(n)}$ is essentially the PSP elicited by a single spike from the nth input neuron; $r_2^{(n)}$ describes the joint nonlinear effect of pairs of spikes from the nth input neuron in addition to the summation of their first-order responses, i.e., $r_1^{(n)}({\tau }_1)+r_1^{(n)}({\tau }_2)$. The term $r_{2x}^{(n_1,n_2)}({\tau }_1,{\tau }_2)$ represents the joint nonlinear effect of pairs of spikes with one spike from neuron $n_1$ and one spike from neuron $n_2$ (Fig. 13.4).

In this formulation, the ensemble neuronal input–output properties are represented by model coefficients $S=[k,h]$. All coefficients are simultaneously estimated from input and output spike trains (x and y). Specifically, the first-order response function $r_1$ can be interpreted as the PSP; the second-order response function $r_2$ is equivalent to the paired-pulse STSP.

13.2.2 Nonlinear Dynamical Model of the Hippocampal CA3-CA1

We have applied nonlinear dynamical modeling to the hippocampal CA3 to CA1 in rodents, nonhuman primates and human subjects [8,10,30–32,57,58,66] and the prefrontal cortex layer 2/3 to layer 5 in nonhuman primates [29,61]. In these applications, the input–output spike trains are recorded from well-trained animals, where the input and output signals as well as the input–output transformations are stabilized and thus sufficiently described by stationary GVMs. Fig. 13.5 shows a GVM of a hippocampal CA1 neuron estimated from a rodent performing a delayed-nonmatch-to-sample (DNMS) task. Results show that this neuron receives inputs from 6 out of 24 CA3 neurons. The single-pulse and paired-pulse response functions ($r_1$ and $r_2$) are the PSPs and paired-pulse facilitation/depression functions inferred from the CA3-CA1 spiking activities. The zero order kernel is −1. The threshold is 0. The standard deviation of the Gaussian noise is estimated to be 0.418. This MISO model can accurately predict the output spike train based on the input spike trains (Fig. 13.5, bottom-middle) as verified with the out-of-sample Kolmogorov–Smirnov test based on the time rescaling theorem (Fig. 13.5, bottom-right). In this stationary model, all model variables are time-invariant.

13.3.1 Theory

The nonlinear dynamical spiking neuron model has provided a quantitative way to infer PSP and STSP from spiking input–output activities. To further infer LTSP, we extend the GVM to be time varying to track system nonstationarity. GVM coefficients are estimated recursively from the input–output spikes and the changes of the nonlinear dynamics are described by the temporal evolution of the kernel functions (Fig. 13.2, middle).

We develop the nonstationary model by combining the GVM and the point-process adaptive filtering method [18,25,65]. GVM coefficients (c) are taken as state variables while the input–output spikes are taken as observable variables. Using adaptive filtering methods, state variables are recursively updated as the observable variables unfold in time. The underlying change of system input–output properties is then represented by the time varying GVM ($S(t)=[k(t),h(t)]$) reconstructed with the time varying coefficients $c(t)$.

Specifically, the probability of observing an output spike at time t, i.e., $P(t)$, is predicted by the GVM at time $t-1$ based on the inputs up to t and output before t. Secondly, the difference between $P(t)$ and the new observation of output $y(t)$ is used to update the GVM coefficients. Using the stochastic state point process filtering algorithm, coefficient vector $C(t)$ and its covariance matrix $W(t)$ are both updated iteratively at each time step t as follows:

$P(t)=0.5-0.5\cdot \mathrm{erf}\left(\frac{\theta -u(t)-a(t)}{\sqrt{2}\sigma }\right),$

$c(t)=c(t-1)+W(t)\left[{\left(\frac{\partial \log P(t)}{\partial C(t)}\right)}^T\right(y(t)-P(t)\left)\right],$

$\begin{array}{c}W{(t)}^{-1}={[W(t-1)+Q]}^{-1}+\hfill \\ [{\left(\frac{\partial \log P(t)}{\partial c(t)}\right)}^{T}P(t)(\frac{\partial \log P(t)}{\partial c(t)})-(y(t)-P(t)\left)\frac{{\partial }^2\log P(t)}{\partial c(t)\partial c{(t)}^{\prime }}\right],\hfill \end{array}$

where Q is the coefficient noise covariance matrix and erf is the Gauss error function. In this method, W acts as the adaptive “learning rate” that allows reliable and rapid tracking of the model coefficients.

13.3.2 Simulation Studies on Nonstationary Nonlinear Dynamical Model

We have tested intensively the nonstationary nonlinear dynamical modeling algorithm with synthetic input–output spike train data obtained through simulations. In all simulations, the inputs are Poisson random spike trains with mean firing rates ranging from 2 Hz to 6 Hz.

13.3.2.1 Step Changes

First, we simulate a second-order, two-input, single-output spiking neuron model with step changes. The total simulation length is 8000 s. At 4000 s, the first-order kernel and the second-order self-kernel of the first input (i.e., $k_1^{(1)}$ and $k_{2s}^{(1)}$) decrease the amplitudes by half with the same waveforms; the first-order kernel and the second-order self kernel of the second input (i.e., $k_1^{(2)}$ and $k_{2s}^{(2)}$) double the amplitudes with the same waveforms. Other kernels (i.e., $k_0$, $k_{2x}^{(1,2)}$, and h) remain constant during the whole simulation. Using the simulated input–output spiking data, we apply the nonstationary nonlinear dynamical modeling algorithm to track the changes of the kernels. Results show that all kernels as well as the changes of the kernels can be recovered accurately from the simulated input–output spike trains (Fig. 13.6). In this nonstationary model, model variables become time varying.

13.3.2.2 LTP and LTD-Like Changes

We further test the capability of the nonstationary modeling algorithm to track biologically plausible forms of synaptic plasticity. A first-order, two-input system with concurrent LTP- and LTD-like changes at different inputs is simulated (Fig. 13.7). The total simulation length is 9000 s. At 3000 s, the kernels of the first input and the second input have LTP-like change (i.e., a near instantaneous increase followed by an exponential decay to a potentiated level double the initial amplitude) and LTD-like change (i.e., an exponential decay to half of the initial amplitude), respectively. Results show that the estimated kernels (red (gray in print version)) track faithfully the actual kernels (black) during the whole simulation, given the LTP-like change is much more challenging than the step change.

13.3.2.3 Input-Independent Changes

In our formulation of the nonstationary model, the zero order time varying kernel $k_0(t)$ (i.e., $c_0(t)$) in Eq. (13.13) describes the input-independent nonstationarity, i.e., changes of the output firing probability caused by latent factors other than the observed input spike trains. To test whether the algorithm can gracefully handle this type of nonstationarity, we modify the step change simulation in Section 13.3.2.1 by varying the input-independent baseline firing probability following a linear chirp (Fig. 13.8, top) and a random noise sequence (Fig. 13.8, bottom). Results show that the estimated baselines (red (gray in print version)) rapidly converge to the actual baselines (black) in both simulations without interfering with the estimation of higher-order kernels, which represent the input-dependent nonstationarities.

13.3.2.4 Larger Number of Inputs

To validate the capability of the algorithm to tackle simultaneous gradual changes with larger number of inputs, we design a first-order, eight-input system with sigmoidal changes. Among the eight inputs, three inputs increase amplitudes following a sigmoidal curve; three inputs decrease amplitudes following a sigmoidal curve; two inputs remain constant. The output mean firing rates before and after the sigmoidal changes are 3.677 Hz and 5.112 Hz, respectively. The total simulation length is 9000 s. Results show that all eight estimated kernels converge to the actual kernels during the simulation (Fig. 13.9).

13.4.1 Theory

The nonstationary GVM above describes how LTSP evolves over time. In the last step, we seek to explain how LTSP is induced by a synaptic learning rule (Fig. 13.2, bottom).

Given the estimated nonstationary model $S(t)$, the change at time t can be calculated as $\mathrm{\Delta }S(t)=S(t)-S(t-1)$, i.e., $\mathrm{\Delta }k(t)=k(t)-k(t-1)$, $\mathrm{\Delta }h(t)=h(t)-h(t-1)$ and $\mathrm{\Delta }S(t)=[\mathrm{\Delta }k(t),\mathrm{\Delta }h(t)]$. Given that $\mathrm{\Delta }S(t)$ is caused by the preceding input and output activities ($z=[x,y]$), the sought ensemble synaptic learning rule L is essentially the causal relationship between the spatiotemporal pattern z and $\mathrm{\Delta }S(t)$ (Fig. 13.10). The $\mathrm{\Delta }S(t)$ can be expressed as a Volterra functional power series of z in a finite memory window ($M_L$) as

$\begin{array}{cc}\hfill \mathrm{\Delta }S(t)& =L_0+\sum _{n=1}^{N+1}\sum _{\tau =0}^{M_L}{L}_1^{(n)}(\tau )z_n(t-\tau )\hfill \\ \hfill & +\sum _{n_1=1}^{N+1}\sum _{n_2=1}^{n_1}\sum _{{\tau }_1=0}^{M_L}\sum _{{\tau }_2=0}^{M_L}{L}_2^{(n_1,n_2)}({\tau }_1,{\tau }_2)z_{n_1}(t-{\tau }_1)z_{n_2}(t-{\tau }_2)\hfill \\ \hfill & +\sum _{n_1=1}^{N+1}\sum _{n_2=1}^{n_1}\sum _{n_3=1}^{n_2}\sum _{{\tau }_1=0}^{M_L}\sum _{{\tau }_2=0}^{M_L}\sum _{{\tau }_3=0}^{M_L}{L}_3^{(n_1,n_2,n_3)}({\tau }_1,{\tau }_2,{\tau }_3)z_{n_1}(t-{\tau }_1)z_{n_2}(t-{\tau }_2)z_{n_3}(t-{\tau }_3)+....\hfill \end{array}$

In this formulation, $L_0$ is the zero order learning rule describing the input/output activity–independent drift of the system; $L_1$ is the first-order learning rule describing the linear relation between the changes of the GVM and the input or output activities; $L_2$ is the second-order learning rule describing how the pair-wise nonlinear interactions between input/output spikes change the GVM; $L_3$ is the third-order learning rule describing how the triplet-wise nonlinear interactions between input/output spikes change the MIMO model (note that there are redundancies in $L_2$ and $L_3$ when $n_1$, $n_2$ and $n_3$ contain the same inputs. Eq. (13.15) is used nonetheless for its simplicity).

Since $\mathrm{\Delta }S(t)$ has been estimated, x and y are known, Eq. (13.15) is essentially a linear model and the learning rule L can thus be estimated with the standard least squares method. In practice, basis functions and penalized likelihood estimation methods can be utilized to reduce the total number of coefficients and select the significant terms in Eq. (13.15). Compared with existing learning rules, such as (a) the classical Bienenstock–Cooper–Munro (BCM) model of synaptic modification and (b) the STDP, this formulation provides a general form of ensemble learning rule defining how the changes of GVM are determined by the spatio-temporal patterns of the input–output spikes. Linear and nonlinear interactions between multiple input/output spikes with different interspike intervals are explicitly included in this formulation (Fig. 13.10).

13.4.2 Relationship Between Volterra Kernels and STDP

Eq. (13.15) provides a general representation of the ensemble learning rule underlying the changes of the neuronal input–output function. It utilizes a Volterra power series to describe how the GVM variables change as a consequence of the interactions between the input–output spikes. Before testing the identification method with simulation, it is instructive to elucidate the relation between the Volterra kernels and the most accepted synaptic learning rule, i.e., STDP (Fig. 13.11).

According to the STDP rule, the change of synaptic weight between a presynaptic neuron and a postsynaptic neuron is determined by the timing of the presynaptic and postsynaptic spiking activities (${\tau }_x$ and ${\tau }_y$). If a presynaptic (i.e., input) spike precedes a postsynaptic spike within a short time window (i.e., ${\tau }_x>{\tau }_y$), the synaptic weight will be enhanced; if the opposite happens (i.e., ${\tau }_x<{\tau }_y$), the synaptic weight will be reduced. The relationship between the amount of synaptic weight change and the spike timings, i.e., the STDP function, can be described with two exponential curves (Fig. 13.11, left). Since in this STDP expression the synaptic weight change depends only on the pair-wise interaction between a presynaptic spike and a postsynaptic spike, the Volterra expression of the learning rule contains only the second-order cross term between x and y. In the case of a single-input system, Eq. (13.15) is reduced to

$\mathrm{\Delta }S(t)=\sum _{{\tau }_x=0}^{M_L}\sum _{{\tau }_y=0}^{M_L}{L}_2^{(x,y)}({\tau }_x,{\tau }_y)x(t-{\tau }_x)y(t-{\tau }_y).$

In $L_2^{(x,y)}$, the STDP function describes only the steady-state level of synaptic weight change. Another factor that contributes to $L_2^{(x,y)}$ is the induction dynamics of the STDP, i.e., the synaptic weight does not change instantaneously following a delta function; instead, it relatively slowly builds up and then stabilizes following a smooth induction function I (Fig. 13.11, right). The STDP function and the induction function I jointly determine the shape of $L_2^{(x,y)}$ (Fig. 13.11, middle-top). In other words, this form of cross kernel between x and y is the Volterra representation of the STDP and induction functions.

13.4.3 Simulation Studies on Learning Rule Identification

We further test the learning rule identification method with simulated input–output spiking data. A first-order, single-input, single-output spiking neuron model is built with the GVM structure (Fig. 13.12). In this model, the first-order feedforward kernel $k_1$ has a typical EPSP shape determined by two exponentials with the time constants at 30 ms and 150 ms, controlling the onset and decay rates, respectively. The peak amplitude is 0.3. The feedback kernel has a negative exponential shape to model the refractory period of spike generation. The time constant and peak amplitude are 10 ms and −1. The noise standard deviation σ is 0.4. During the simulation, the input spike train x is fed into the model to generate the output spike train y. The synaptic strength between the input and output neurons, i.e., $k_1$, is changed following the standard STDP rule and the induction function. In this simulation, the shape of $k_1$ remains the same; only the amplitude is changed. For convenience, we use $k_1$ to represent both the peak amplitude and the whole kernel function. The left (LTD) half of the STDP function is a single exponential with time constant and peak amplitude at 33.7 ms and −0.018. The right (LTP) half of the STDP function is a single exponential with time constant and peak amplitude at 16.8 ms and 0.032. Both sides share the same induction function determined by two exponentials with time constants at 900 ms and 4500 ms, for onset and decay rates, respectively. Without loss of generality, the integral of the induction function is set at 1. To prevent the intrinsic instability of the simulated system, we adjust the input patterns based on the level of $k_1$. When $k_1$ reaches a value below a certain threshold, the input x changes from a 5-Hz Poisson random train to a 50-Hz burst to cause more potentiation and increase of $k_1$; when $k_1$ is above the threshold, x is shifted to spike within 10 ms after output spikes to cause more depression and decrease of $k_1$. This simple method ensures $k_1$ to fluctuate in a stable manner. The total length of simulation is 200 s.

Using the nonstationary GVM method, we first track the changes of the synaptic strength, i.e., $k_1(t)$, and compare them with the actual values. Results show that $k_1(t)$ can be accurately recovered from the simulated input–output spike trains (Fig. 13.13, top).

Given $k_1(t)$, we further estimate the learning rule, i.e., the second-order cross kernel $L_2^{(x,y)}$. In order to reduce the number of open parameters, we expand the kernel with three sets of Laguerre basis functions as

$\mathrm{\Delta }{k}_1(t)=\sum _{j_A=1}^{L_A}\sum _{j_{\psi }=1}^{L_{\psi }}{c}_{xy}(j_A,j_{\psi })v_{xy}^{j_A}(t)v_{xy}^{j_{\psi }}(t)+\sum _{j_A=1}^{L_A}\sum _{j_{\psi }=1}^{L_{\psi }}{c}_{yx}(j_A,j_{\psi })v_{yx}^{j_A}(t)v_{yx}^{j_{\psi }}(t),$

where

$v_{xy}^{j_A}(t)=\sum _{{\tau }_x=1}^{M_{\psi }}{b}_{xy}^{j_A}({\tau }_x-{\tau }_y)x(t-{\tau }_x),$

$v_{xy}^{j_{\psi }}(t)=\sum _{{\tau }_y=0}^{{\tau }_x-1}{b}_{xy}^{j_{\psi }}({\tau }_y)y(t-{\tau }_y),$

$v_{yx}^{j_A}(t)=\sum _{{\tau }_y=1}^{M_{\psi }}{b}_{yx}^{j_A}({\tau }_y-{\tau }_x)y(t-{\tau }_y),$

$v_{yx}^{j_{\psi }}(t)=\sum _{{\tau }_x=0}^{{\tau }_y-1}{b}_{yx}^{j_{\psi }}({\tau }_y)x(t-{\tau }_x).$

In these equations, c represents the sought-after learning rule coefficients. They are split into $c_{xy}$ and $c_{yx}$ to represent the two halves of the cross kernel for x preceding y and y preceding x, respectively. The subscript A represents the STDP amplitude. The subscript ψ represents the STDP induction. Since all v can be calculated with the predefined basis functions and the known x and y, Eq. (13.17) is essentially linear and c can be estimated with a least squares method. With the estimated coefficients, $L_2^{(x,y)}$ can be reconstructed as

$\begin{array}{cc}\hfill L_2^{(x,y)}({\tau }_x,{\tau }_y)& =\sum _{j_A=1}^{L_A}\sum _{j_{\psi }=1}^{L_{\psi }}{c}_{xy}(j_A,j_{\psi })b_{xy}^{j_A}({\tau }_x-{\tau }_y)b_{xy}^{j_{\psi }}({\tau }_y)\hfill \\ \hfill & +\sum _{j_A=1}^{L_A}\sum _{j_{\psi }=1}^{L_{\psi }}{c}_{yx}(j_A,j_{\psi })b_{yx}^{j_A}({\tau }_y-{\tau }_x)b_{yx}^{j_{\psi }}({\tau }_x).\hfill \end{array}$

Here, $L_2^{(x,y)}$ constitutes a general representation of the synaptic learning since it does not require the independence of the STDP function and the induction function. In this study, however, since these two functions are assumed to be independent, $L_2^{(x,y)}$ is equal to the outer product of the two functions. The left (LTD) side of the STDP function can be obtained by integrating the vertical vectors in the upper half of $L_2^{(x,y)}$ along its diagonal line; similarly, the right (LTP) side of the STDP function can be obtained by integrating the horizontal vectors in the lower half of $L_2^{(x,y)}$ along its diagonal line. The induction function for LTD can be obtained by integrating the diagonal vectors in the upper half of $L_2^{(x,y)}$ along the vertical axis; similarly, the induction function for LTP can be obtained by integrating the diagonal vectors in the lower half of $L_2^{(x,y)}$ along the horizontal axis. The induction function is then normalized to have a unitary integral. The STDP functions are scaled accordingly.

Fig. 13.13 shows the final identification results using the same 200-s input–output spiking data. It is evident that both the STDP function and the induction function are faithfully recovered with this method (Fig. 13.13B, C).