Journal of Computational Neuroscience

, Volume 25, Issue 3, pp 401–438

Theoretical analysis of reverse-time correlation for idealized orientation tuning dynamics

Authors

    • Rensselaer Polytechnic Institute
  • Louis Tao
    • New Jersey Institute of Technology
    • Center for Bioinformatics, School of Life SciencesPeking University
  • David Cai
    • Courant Institute of Mathematical SciencesNew York University
  • Michael J. Shelley
    • Courant Institute of Mathematical SciencesNew York University
Article

DOI: 10.1007/s10827-008-0085-7

Cite this article as:
Kovačič, G., Tao, L., Cai, D. et al. J Comput Neurosci (2008) 25: 401. doi:10.1007/s10827-008-0085-7

Abstract

A theoretical analysis is presented of a reverse-time correlation method used in experimentally investigating orientation tuning dynamics of neurons in the primary visual cortex. An exact mathematical characterization of the method is developed, and its connection with the Volterra–Wiener nonlinear systems theory is described. Various mathematical consequences and possible physiological implications of this analysis are illustrated using exactly solvable idealized models of orientation tuning.

Keywords

Random walksMexican hatPrimary visual cortexOrientation tuning dynamics

Supplementary material

$$ R(z)=\int_0^1 P(z-x) dx~, $$$$ \tilde{R}(\tilde{z})=\sum_{k=-\infty}^{\infty} R(\tilde{z}+k)=\int_{-\infty}^{+\infty} P(x)dx\equiv 1~, $$$$ \left\langle s_0(t_1) s_0(t_2)\right\rangle = A^2\delta(t_1-t_2), $$$$ \left\langle s_0(t_1) \ldots s_0(t_{2n+1})\right\rangle = 0, $$$$ \left\langle s_0(t_1) \ldots s_0(t_{2n})\right\rangle = \sum\prod \left\langle s_0(t_i) s_0(t_j)\right\rangle. $$$$ \begin{array}{lll}\label{walk} s_\nu(t)&=&\frac{A}{\sqrt\nu}\left\{\begin{array}{ll}p,\quad &\mbox{with probability } q,\\[10pt] -q,\quad &\mbox{with probability } p,\end{array}\right.\nonumber\\ &&{\kern22.6pt} k\nu< t < (k+1)\nu,\quad k=\ldots,-1,0,1,\ldots,\nonumber\\ \end{array} $$$$\label{singleavg} \langle s_\nu(t)\rangle =0. $$$$\label{boxcorrelations} \langle s_\nu(t_1)\ldots s_\nu(t_m)\rangle =A^m\nu^{-\frac{m}{2}} \left[p^mq +p (-q)^m\right]. $$$$\label{nucorrelations} \langle s_\nu(t_1)\ldots s_\nu(t_n)\rangle=\sum\prod \langle s_\nu(t_i)\ldots s_\nu(t_j)\rangle, $$$$\label{showdelta} \int_{-\infty}^\infty\ldots \int_{-\infty}^\infty L(\tau_1,\ldots,\tau_n) \left\langle s(t-\tau_1)\ldots s(t-\tau_n)\right\rangle \,d\tau_1 \ldots d\tau_n, $$$$\label{prodint} \int_{-\infty}^\infty\!\ldots\! \int_{-\infty}^\infty\!\! L(\tau_1,\ldots,\tau_n)\!\prod_{j=1}^l \!\left\langle s\left(t\!-\!\tau_{p_{j-1}+1}\right)\ldots s\left(t\!-\!\tau_{p_j}\right)\right\rangle d\tau_1 \ldots d\tau_n $$$$ \begin{array}{lll}\label{sumandint} &&{\kern-6pt} A^n \nu^{-\frac{n}{2}}\sum\limits_{m_1=-\infty}^\infty \ldots \sum\limits_{m_l=-\infty}^\infty\int_0^\nu\ldots \int_0^\nu L(t-m_1\nu-\sigma_1, \ldots,t-m_1\nu-\sigma_{p_1},\nonumber \\[2pt]&&\qquad\qquad\qquad\qquad\qquad\qquad\quad \ldots,t-m_l\nu-\sigma_{p_{l-1}+1},\ldots,t-m_l\nu-\sigma_{p_l}) \,d\sigma_1 \ldots d\sigma_n \nonumber \\[2pt]&&{\kern11pt} =A^n\nu^{-\frac{n}{2}}\nu^{n-l}\left[\sum\limits_{m_1=-\infty}^\infty \ldots \sum\limits_{m_l=-\infty}^\infty L(t-m_1\nu, \ldots,t-m_1\nu,\ldots,t-m_l\nu,\ldots,t-m_l\nu)\nu^l +{\mathcal O}(\nu)\right] \nonumber \\[2pt]&&{\kern11pt} =A^n\nu^{\frac{n}{2}-l}\left[\int_{-\infty}^\infty\ldots\int_{-\infty}^\infty L(t_1, \ldots,t_1,\ldots,t_l,\ldots,t_l)\,dt_1\ldots dt_l +{\mathcal O}(\nu)\right]. \end{array} $$$$ \begin{array}{lll}\label{differenceofintegrals} &&{\kern-6pt} \nu^{l-n}\!\!\sum\limits_{m_1=-\infty}^\infty \!\!\ldots\!\! \sum\limits_{m_l=-\infty}^\infty\!\int_0^\nu\!\!\ldots\!\! \int_0^\nu\!\! L(t\!-\!m_1\nu\!-\!\sigma_1, \ldots,t\!-\!m_1\nu-\sigma_{p_1},\nonumber \\[2pt]&& {\kern5pt} \ldots,t-m_l\nu-\sigma_{p_{l-1}+1},\ldots,t-m_l\nu-\sigma_{p_l}) \,d\sigma_1 \ldots d\sigma_n \nonumber \\[2pt] &&{\kern5pt} -\int_{-\infty}^\infty\ldots\int_{-\infty}^\infty L(t_1, \ldots,t_1,\ldots,t_l,\ldots,t_l)\,dt_1\ldots dt_l, \end{array} $$$$ \begin{array}{lll} &&{\kern-6pt} \nu^{l-n}\!\sum\limits_{m_1=-\infty}^\infty \!\!\ldots\!\! \sum\limits_{m_l=-\infty}^\infty\!\int_0^\nu\!\!\ldots\!\! \int_0^\nu \Bigl[\! L(t\!-\!m_1\nu\!-\!\sigma_1, \ldots,t\!-\!m_1\nu\!-\!\sigma_{p_1},\nonumber \\[6pt]&&{\kern12pt} \ldots,t-m_l\nu-\sigma_{p_{l-1}+1},\ldots,t-m_l\nu-\sigma_{p_l})\nonumber \\[2pt] &&{\kern5pt} -L(t-m_1\nu-\sigma_{p_1}, \ldots,t-m_1\nu-\sigma_{p_1},\nonumber \\[2pt]&&{\kern12pt} \ldots,t-m_l\nu-\sigma_{p_l},\ldots,t-m_l\nu-\sigma_{p_l}) \Bigr]\,d\sigma_1 \ldots d\sigma_n. \end{array} $$$$ \begin{array}{lll} &&{\kern-6pt} R\big(t-m_1\nu-\sigma_{p_1}, t-m_2\nu-\sigma_{p_2},\ldots,t-m_l\nu-\sigma_{p_l}\big)\nonumber\\[3pt] &&{\kern6pt}\times \bigl( |\sigma_1-\sigma_{p_1}|+\ldots+|\sigma_{p_1-1}-\sigma_{p_1}| +|\sigma_{p_1+1}-\sigma_{p_2}|+\nonumber \\[3pt] &&{\kern19pt} \ldots + |\sigma_{p_2-1}-\sigma_{p_2}|+\ldots + |\sigma_{p_{l-1}+1}-\sigma_{p_l}|+\nonumber\\[3pt] &&{\kern19pt} \ldots +|\sigma_{p_l-1}-\sigma_{p_l}|\bigr), \end{array} $$$$ (t-m_1\nu-\sigma_{p_1}, \ldots,t-m_1\nu-\sigma_{p_1},\ldots,t-m_l\nu-\sigma_{p_l},\ldots,t-m_l\nu-\sigma_{p_l}) $$$$ \int_0^\nu |\sigma_i-\sigma_j|\,d\sigma_i=\frac{\nu^2}{4}+\left(\frac{\nu}{2}-\sigma_j\right)^2, $$$$ \begin{array}{lll} &&\nu^{l-n}(n-l)\frac{\nu^{n+1-l}}{2} \sum\limits_{m_1=-\infty}^\infty \ldots \sum\limits_{m_l=-\infty}^\infty\int_0^\nu\ldots\nonumber\\&&{\kern12pt} \int_0^\nu R(t-m_1\nu-u_1,\ldots,t-m_l\nu-u_l)\,du_1\ldots du_l\nonumber \\[12pt]&&{\kern20pt} =\frac{(n-l)\nu}{2}\int_{-\infty}^\infty\ldots \int_{-\infty}^\infty R(v_1,\ldots,v_l)\,dv_1\ldots dv_l \nonumber\\[12pt]&&{\kern20pt} = {\mathcal O}(\nu), \end{array} $$$$\label{wienerseries} r[s](t)\!=\!P_0[s]\! +\!P_1[s](t)\!+\!P_2[s](t)\!+\cdots +\! P_n[s](t)\! + \cdots, $$$$\label{kernels} W_m(T_1,\ldots,T_m)= \frac{1}{A^{2m} m!}\left\langle r[s_0](t) s_0(t-T_1)\ldots s_0(t-T_m)\right\rangle , $$$$ \begin{array}{lll}\label{kernels1} &&{\kern-6pt} W_m(T_1,\ldots,T_m)\nonumber\\&&= \frac{1}{A^{2m} m!} \lim\limits_{T\to\infty} \frac{1}{2T}\int_{-T}^T r[s_0](t) s_0(t-T_1)\ldots s_0(t-T_m)\,ds\nonumber\\\end{array}$$$$ \begin{array}{lll}\label{kernels2} &&{\kern-6pt} W_m(T_1,\ldots,T_m)\nonumber\\&& =\! \frac{1}{A^{2m} m!}\left\langle \left\{r[s_0](t)\!-\!r_{m-1}[s_0](t)\right\} s_0(t\!-\!T_1)\ldots s_0(t\!-\!T_m)\right\rangle ,\nonumber\\ \end{array} $$$$ P_1[s](t) = \int_0^\infty W_1(T_1) s(t-T_1)\,dT_1, $$$$ \begin{array}{lll}\label{probe1} &&{\kern-6pt} \tilde W_m(T_1,\ldots,T_m)\nonumber\\ &&{\kern6pt} =\left\langle r[s_\nu](t)s_\nu(t\!-\!T_1)\ldots s_\nu(t\!-\!T_m)\right\rangle \nonumber \\[2pt] &&{\kern6pt} = L_0 \left\langle s_\nu(t\!-\!T_1)\ldots s_\nu(t\!-\!T_m)\right\rangle\nonumber\\ &&{\kern18pt} +\int_0^\infty L_1(\tau_1) \left\langle s_\nu(t\!-\!T_1)\ldots s_\nu(t\!-\!T_m)s_\nu(t\!-\!\tau_1)\right\rangle\,d\tau_1\nonumber \\[2pt]&&{\kern18pt} +\frac{1}{2!}\int_0^\infty \int_0^\infty L_2(\tau_1,\tau_2) \big\langle s_\nu(t\!-\!T_1)\ldots s_\nu(t\!-\!T_m)\nonumber\\ &&{\kern18pt}\times s_\nu(t\!-\!\tau_1)s_\nu(t\!-\!\tau_2)\big\rangle\,d\tau_1d\tau_2 +\ldots \nonumber \\[2pt]&&{\kern18pt} + \frac{1}{n!}\int_0^\infty\!\!\ldots\! \int_0^\infty\!\! L_n(\tau_1,\ldots,\tau_n)\left\langle s_\nu(t\!-\!T_1)\ldots s_\nu(t\!-\!T_m) \right.\nonumber \\[2pt]&&{\kern18pt} \left. \times s_\nu(t\!-\!\tau_1)\ldots s_\nu(t\!-\!\tau_n)\right\rangle \,d\tau_1 \ldots d\tau_n + \ldots. \nonumber\\ \end{array} $$$$ \begin{array}{lll}&&{\kern-7pt}  \left\langle s_\nu(t-T_1)\ldots s_\nu(t-T_{q_1})\right\rangle \ldots\left\langle s_\nu(t-T_{q_{i-1}+1})\ldots s_\nu(t-T_{q_i})\right\rangle\nonumber \\[6pt]&&{\kern.7pc}  \int_{-\infty}^\infty\ldots \int_{-\infty}^\infty L_n(\tau_1,\ldots,\tau_n)\big\langle s_\nu(t-\tau_1) \ldots s_\nu(t-\tau_{p_1})\big\rangle \ldots\nonumber \\[6pt]&&{\kern40pt}  \times\left\langle s_\nu(t\!-\!\tau_{p_{j\!-\!1}+1})\ldots s_\nu(t\!-\!\tau_{p_j})\right\rangle \big\langle s_\nu(t\!-\!T_{q_i+1})\nonumber \ldots\\[6pt]&&{\kern40pt}  \times s_\nu(t\!-\!T_{q_{i+1}})s_\nu(t\!-\!\tau_{p_j+1})\ldots s_\nu(t\!-\!\tau_{p_{j+1}})\big\rangle\dots\nonumber \\[6pt]&& {\kern40pt}  \times \big\langle s_\nu(t\!-\!T_{q_{h-1}+1})\ldots s_\nu(t-T_{q_h})s_\nu(t-\tau_{p_{k-1}+1})\ldots\nonumber\\&&{\kern40pt} \times s_\nu(t-\tau_{p_k})\big\rangle d\tau_1\ldots\, d\tau_n. \end{array} $$$$ \begin{array}{lll}\label{nwin} &&{\kern-6pt} \nu^{-\frac{n+m}{2}}B(T_1,\ldots,T_{q_1})\ldots B(T_{q_{i-1}+1},\ldots,T_{q_i})B(T_{q_i+1},\ldots,T_{q_{i+1}})\ldots B(T_{q_{h-1}+1},\ldots,T_{q_h})\nonumber \\[3pt] &&{\kern2pt} \times\sum\limits_{m_1=-\infty}^\infty\ldots \sum\limits_{m_i=-\infty}^\infty\int_0^\nu\ldots\int_0^\nu L_n(t-m_1\nu-\sigma_1, \ldots, t-m_1\nu-\sigma_{p_1},\ldots,\nonumber \\[3pt] &&{\kern6pt} t-m_i\nu-\sigma_{p_{i-1}+1}, \ldots, t-m_i\nu-\sigma_{p_i}, t-k_1\nu-\sigma_{p_i+1},\ldots,t-k_1\nu-\sigma_{p_{i+1}},\ldots ,\nonumber \\[3pt] &&{\kern6pt} t-k_l\nu-\sigma_{p_{k-1}+1},\ldots,t-k_l\nu-\sigma_{p_k})\,d\sigma_1\ldots d\sigma_n \nonumber \\ &&{\kern18pt} =\nu^{-\frac{n+m}{2}}B(T_1,\ldots,T_{q_1})\ldots B(T_{q_{i-1}+1},\ldots,T_{q_i})B(T_{q_i+1},\ldots,T_{q_{i+1}})\ldots B(T_{q_{h-1}+1},\ldots,T_{q_h})\nonumber \\[3pt] &&{\kern30pt} \times\nu^{p_j-j}\nu^{n-p_j}\Biggl[ \int_{-\infty}^\infty\ldots \int_{-\infty}^\infty L_n(u_1,\ldots,u_1,\ldots,u_j,\ldots u_j,t\!-\!k_1\nu,\ldots,t\!-\!k_1\nu,\ldots\nonumber\\[3pt]&&{\kern90pt} \ldots,t\!-\!k_l\nu,\ldots,t\!-\!k_l\nu)\,du_1\ldots du_j \!+\!{\mathcal O}(\nu)\!\Biggr] \nonumber \\ &&{\kern18pt} =\nu^{\frac{n-m}{2}-j}B(T_1,\ldots,T_{q_1})\ldots B(T_{q_{i-1}+1},\ldots,T_{q_i})B(T_{q_i+1},\ldots,T_{q_{i+1}})\ldots B(T_{q_{h-1}+1},\ldots,T_{q_h}) \Biggl[ \int_{-\infty}^\infty\ldots \int_{-\infty}^\infty \nonumber \\[3pt] &&{\kern30pt} \times\! L_n(u_1,\ldots,u_1,\ldots,u_j,\ldots u_j,t-k_1\nu,\ldots,t-k_1\nu,\ldots,t-k_l\nu,\ldots,t-k_l\nu)\,du_1\ldots du_j + {\mathcal O}(\nu) \Biggr]. \end{array} $$$$ \begin{array}{lll}&&{\kern-6pt} k_1\nu<T_{q_i+1},\ldots,T_{q_{i+1}}<(k_1+1)\nu.\\&&{\kern-6pt} \ldots\\&&{\kern-6pt} k_l\nu<T_{q_{h-1}+1},\ldots,T_{q_h}<(k_l+1)\nu. \end{array}$$$$ \label{flashchar} {\mathcal I}({\bf X},t) =\!{\mathcal I}_0\nu^{-1/2}\left\{1\!+\!\varepsilon \sum\limits_{i=1}^I\sum\limits_{j=1}^J\chi_{\theta_i,\phi_j}(t)\sin[{\bf k}(\theta_i)\cdot{\bf X}\!-\!\phi_j]\right\}, $$$$ \label{flashwalk} {\mathcal I}({\bf X},t) =\!{\mathcal I}_0\!\left\{\!\nu^{-1/2}\!+\!\varepsilon \sum\limits_{i=1}^I\sum\limits_{j=1}^J \left[s_{\nu i j}(t)\!+\!\frac{1}{\sqrt{\nu}\, I J} \right]\sin[{\bf k}(\theta_i)\cdot{\bf X}-\phi_j]\!\right\}, $$$$ s_{\nu i j}(t)\!=\!\nu^{-1/2}\left[\!\chi_{\theta_i,\phi_j}(t)\!-\!\frac{1}{IJ}\!\right],\;\, i\!=\!1,\ldots,I,\;\, j\!=\!1,\ldots,J, $$$$ \begin{array}{lll}\label{onewalk} s_{\nu ij}(t)&=&\frac{1}{\sqrt\nu}\left\{\begin{array}{ll} p,\quad &\mbox{with probability } q,\\[4pt] -q,\quad &\mbox{with probability } p,\\[4pt] \end{array}\right. \nonumber\\[2pt] && k\nu< t < (k+1)\nu, \quad k=\ldots,-1,0,1,\ldots,\nonumber\\ \end{array} $$$$q=\frac{1}{IJ}, \qquad p=1-\frac{1}{IJ}.$$$$\label{secondwalk} s_{\nu mn}(t)=\frac{1}{\sqrt\nu}\left\{\begin{array}{ll} p,\quad &\mbox{with probability } \displaystyle \frac{1}{IJ-1},\\[10pt] -q,\quad &\mbox{with probability } \displaystyle 1- \frac{1}{IJ-1},\end{array}\right. $$$$\label{zeromean1} \left\langle s_{ij} (t)\right\rangle_{\vec s}=0, $$$$ \left\langle s_{ij}(t_1) s_{ij}(t_2)\right\rangle_{\vec s} = \frac{IJ-1}{I^2J^2}\; \delta(t_1-t_2), $$$$ \left\langle s_{ij}(t_1) s_{kl}(t_2)\right\rangle_{\vec s} = -\frac{1}{I^2J^2}\; \delta(t_1-t_2), \qquad i\neq k \mbox{ or } j\neq l, $$$$ \left\langle s_{i_1j_1}(t_1) \ldots s_{i_{2n+1}j_{2n+1}}(t_{2n+1})\right\rangle_{\vec s} = 0, $$$$ \left\langle s_{i_1j_1}(t_1) \ldots s_{i_{2n}j_{2n}}(t_{2n})\right\rangle_{\vec s} = \sum\prod \left\langle s_{i_lj_l}(t_l) s_{i_mj_m}(t_m)\right\rangle_{\vec s}. $$$$\sum\limits_{i=1}^I\sum\limits_{j=1}^J s_{ij}(t) = 0. $$$$ p_0[\vec S](t) = w_0,$$$$p_1[\vec S](t)= \sum\limits_{i=1}^I \sum\limits_{j=1}^J \int_0^\infty w_1^{ij}(\tau_1) S_{ij}(t-\tau_1)\,d\tau_1 + w_{1,0}$$$$\label{firstfunc} p_1[\vec S](t)= \sum\limits_{i=1}^I \sum\limits_{j=1}^J \int_0^\infty w_1^{ij}(\tau_1) S_{ij}(t-\tau_1)\,d\tau_1.$$$$\label{zerosumker} \sum\limits_{i=1}^I \sum\limits_{j=1}^J w_1^{ij}(\tau_1)=0.$$$$ \begin{array}{lll}p_2[\vec S](t)\!&=&\!\sum\limits_{i,k=1}^I\sum\limits_{j,l=1}^J \!\int_0^\infty \!\!\!\int_0^\infty \!\!w_2^{ik,jl}(\tau_1,\tau_2)S_{ij}(t\!-\!\tau_1)S_{kl}(t\!-\!\tau_2)\,d\tau_1 d\tau_2\\[6pt] && -\frac{1}{IJ}\sum\limits_{i=1}^I\sum\limits_{j=1}^J \int_0^\infty w_2^{ii,jj}(\tau_1,\tau_1)\,d\tau_1\\[6pt] &&+\frac{1}{I^2J^2} \sum\limits_{i,k=1}^I\sum\limits_{j,l=1}^J \int_0^\infty w_2^{ik,jl}(\tau_1,\tau_1)\,d\tau_1 . \end{array} $$$$\sum\limits_{i=1}^I\sum\limits_{j=1}^J w_2^{ik,jl}(\tau_1,\tau_2)= \sum\limits_{k=1}^I\sum\limits_{l=1}^J w_2^{ik,jl}(\tau_1,\tau_2)=0, $$$$ \begin{array}{lll}p_2[\vec S](t)\!&=&\!\sum\limits_{i,k=1}^I\!\sum\limits_{j,l=1}^J \!\int_0^\infty \!\!\int_0^\infty\!\! w_2^{ik,jl}(\tau_1,\tau_2)S_{ij}(t\!-\!\tau_1)S_{kl}(t\!-\!\tau_2)\,d\tau_1 d\tau_2\\[4pt] &&-\frac{1}{IJ}\sum\limits_{i=1}^I\sum\limits_{j=1}^J \int_0^\infty w_2^{ii,jj}(\tau_1,\tau_1)\,d\tau_1 . \end{array}$$$$ \begin{array}{lll}\label{ratecorrel} &&{\kern-6pt} \langle m\left[{\vec s}\,\right](t,\Theta,\Phi)s_{ij} (t-\tau)\rangle_{\vec s}\nonumber\\[6pt]&&{\kern6pt} =\sum\limits_{k=1}^I\sum\limits_{l=1}^J \int_0^\infty W_1^{k,l}(\tau_1,\Theta,\Phi)s_{kl}(t-\tau_1)s_{ij}(t-\tau)\,d\tau_1\nonumber\\[6pt] &&{\kern6pt} =\frac{1}{IJ} W_1^{i,j}(\tau,\Theta,\Phi) - \frac{1}{(IJ)^2}\sum\limits_{k=1}^I\sum\limits_{l=1}^J W_1^{k,l}(\tau,\Theta,\Phi).\nonumber\\ \par \end{array} $$$$\sum\limits_{k=1}^I\sum\limits_{l=1}^J W_1^{k,l}(\tau,\Theta,\Phi)=0,$$$$\label{1stker} W_1^{i,j}(\tau,\Theta,\Phi) = IJ\left\langle m\left[{\vec s}\,\right](t,\Theta,\Phi)s_{ij}(t-\tau)\right\rangle_{\vec s}, $$$$ \begin{array}{lll}\label{1stker1} W_1^{i,j}(\tau,\Theta,\Phi) &=& IJ\lim\limits_{T\to\infty}\frac{1}{2T}\int_{-T}^T m\left[{\vec s}\,\right](t,\Theta,\Phi)\notag\\ &&{\kern54pt} \times s_{ij}(t-\tau)\, dt \end{array} $$$$ \begin{array}{lll}\label{1stker2} W_1^{i,j}(\tau,\Theta,\Phi) &\!=\!& IJ\lim\limits_{T\to\infty}\lim\limits_{\nu\to 0}\frac{1}{2T\sqrt\nu}\!\int_{-T}^T m\!\left(t,\Theta,\Phi,\vec\theta,\vec\phi\right)\nonumber\\ &&\times\!\left[\chi_{\theta_i ,\phi_j}(t\!-\!\tau)\!-\!\frac{1}{IJ}\right]\, dt, \end{array} $$$${\mathcal N}(\tau,\theta_i,\phi_j) \!=\!\lim\limits_{T\to\infty} \frac{1}{2T}\int_{-T}^T \chi_{\theta_i,\phi_j}(t\!-\!\tau)\, m (t,\Theta\!,\Phi\!,\vec\theta\!,\vec\phi)\,dt,$$$$ \begin{array}{lll}\label{1stker33} W_1^{i,j}(\tau,\Theta,\Phi) & = &\lim\limits_{\nu\to 0}\frac{1}{\sqrt\nu}\Bigl[IJ{\mathcal N}(\tau,\theta_i,\phi_j)-\bar m (\Theta,\Phi)\Bigr]\nonumber\\[12pt] & = &\lim\limits_{\nu\to 0}\frac{1}{\sqrt\nu}\Bigl[ N(\tau,\theta_i,\phi_j)-\langle N\rangle_{\tau,\theta^{(0)},\phi^{(0)}}\Bigr],\nonumber\\ \end{array} $$$$\label{allaverages}\bar m = \overline {\langle m\rangle_{\vec\theta,\vec\phi}},$$$${\mathcal M}(\tau,\theta_i)=\sum\limits_{j=1}^J {\mathcal N}(\tau,\theta_i,\phi_j).$$$$ \begin{array}{lll}\label{1stkeravg2} W_1^i(\tau,\Theta)&=&\frac{1}{J}\sum\limits_{j=1}^J W_1^{i,j}(\tau,\Theta,\Phi)\nonumber \\[6pt]&=& \lim\limits_{\nu\to 0}\frac{1}{\sqrt\nu}\Bigl[I{\mathcal M}(\tau,\theta_i)-\bar m(\Theta,\Phi)\Bigr]\nonumber\\[6pt]& =&\lim\limits_{\nu\to 0}\frac{1}{\sqrt\nu}\Bigl[M(\tau,\theta_i)-\bar m(\Theta,\Phi)\Bigr] \nonumber\\[6pt]& =&\lim\limits_{\nu\to 0}\frac{1}{\sqrt\nu}\Bigl[M(\tau,\theta_i)-\langle M\rangle_{\tau,\theta^{(0)}}\Bigr], \end{array} $$$$\label{rv1} S=\nu^{-1/2}\sum\limits_{n=-\infty}^\infty f(\theta^{(n)})\,{\mathcal G}(t-n\nu)\cos\phi^{(n)}, $$$$S=\sum\limits_{n=-\infty}^\infty S^{(n)},$$$$S^{(n)}=\nu^{-1/2}f(\theta^{(n)})\,{\mathcal G}(t-n\nu)\cos\phi^{(n)},\label{rvn1}$$$$\label{cosdist} p_{\!X^{(n)}}(x)= \left\{ \begin{array}{ll} \displaystyle \frac{1}{\pi\sqrt{1-x^2}}, \quad& x^2<1, \\[18pt] 0, & x^2\geq 1 . \end{array}\right. $$$$ p_{\!AB}(z)=\int_0^\infty p_A\left(\frac{z}{y}\right)p_B(y)\frac{dy}{y}, $$$$\label{one_term} p_{S^{(n)}}(z)=\frac{1}{\pi}\int_0^\pi p_{X^{(n)}}\left(\frac{z}{\nu^{-1/2}f(\theta^{(n)}){\mathcal G}(t-n\nu)}\right) \times\frac{d\theta^{(n)}}{\nu^{-1/2}f(\theta^{(n)})|{\mathcal G}(t-n\nu)|}. $$$$p_{\theta^{(n)}}(\theta)=\left\{\begin{array}{ll}\displaystyle \frac{1}{\pi},\quad &0\leq \theta<\pi,\\[12pt] 0, & \mbox{otherwise,}\end{array}\right.$$$$ \Phi_n(\kappa)\!=\!\frac{1}{\pi}\!\int_0^\pi\!\! d\theta^{(n)} \!\int_{-\infty}^\infty\!\! du\,e^{i\kappa u \nu^{-1/2} f(\theta^{(n)}){\mathcal G}(t-n\nu)} p_{X^{(n)}}(u). $$$$\frac{2}{\pi}\int_0^1 \frac{\cos\alpha u\;du}{\sqrt{1-u^2}}=J_0(\alpha),$$$$\label{chartwo} \Phi_n(\kappa)\!=\!\frac{1}{\pi}\!\int_0^\pi J_0\Bigl(\kappa \nu^{-1/2} f(\theta^{(n)}){\mathcal G}(t\!-\!n\nu)\Bigr)\; d\theta^{(n)}, \:\: n\!\neq\! 0.$$$$ \Phi_0(\kappa)= J_0\Bigl(\kappa\nu^{-1/2} f(\theta^{(0)}){\mathcal G}(t)\Bigr). $$$$ \begin{array}{lll}\label{charfun} \Phi_S(\kappa)&=&J_0\Bigl(\kappa \nu^{-1/2} f(\theta^{(0)}){\mathcal G}(t)\Bigr)\nonumber \\[-3pt]&& \times\prod\limits_{n=-\infty\atop n\neq 0}^\infty\Biggl\{ \frac{1}{\pi}\int_0^\pi J_0\Bigl(\kappa \nu^{-1/2} f(\theta^{(n)}){\mathcal G}(t-n\nu)\Bigr)\; d\theta^{(n)}\Biggr\}. \nonumber\\[-3pt] \end{array} $$$$ \begin{array}{lll}\label{distrib} &&{\kern-6pt} p_S(s,t,\theta^{(0)})\nonumber \\[-3pt]&&{\kern4pt} = \frac{1}{2\pi}\int_{-\infty}^\infty d\kappa\, e^{-i\kappa s} J_0\Bigl(\kappa \nu^{-1/2} f(\theta^{(0)}){\mathcal G}(t)\Bigr)\nonumber\\[-3pt]&&{\kern15pt} \times\!\prod\limits_{n=-\infty\atop n\neq 0}^\infty\Biggl\{ \frac{1}{\pi}\int_0^\pi\! J_0\Bigl(\kappa \nu^{-1/2} f(\theta^{(n)}){\mathcal G}(t\!-\!n\nu)\Bigr)\; d\theta^{(n)}\Biggr\}.\nonumber\\ \end{array} $$$$\label{sigma} \Sigma = \nu^{-1/2}\sum\limits_{n=-\infty}^\infty f(\theta^{(n)}){\mathcal G}(t-n\nu)\cos\phi^{(n)}, $$$$ \begin{array}{lll}\label{psigma} p_\Sigma (s,t)&\!=\!&\frac{1}{2\pi}\int_{-\infty}^\infty d\kappa\, e^{-i\kappa s} \nonumber\\[-3pt] &&\times\!\prod\limits_{n=-\infty}^\infty\Biggl\{ \frac{1}{\pi}\int_0^\pi\! J_0\Bigl(\kappa \nu^{-1/2} f(\theta^{(n)}){\mathcal G}(t-n\nu)\Bigr) d\theta^{(n)}\Biggr\}, \nonumber\\[-3pt] \end{array} $$$$J_0(x)=\sum\limits_{l=0}^\infty \frac{(-1)^l \, x^{2l}}{2^{2l}(l!)^2}$$$$ J_0\Bigl(\kappa \nu^{-1/2} f(\theta^{(0)}){\mathcal G}(t)\Bigr) =1 - \frac{\kappa^2 f^2(\theta^{(0)})\,{\mathcal G}^2(t)}{4\nu} + \frac{\kappa^4 f^4(\theta^{(0)})\,{\mathcal G}^4(t)}{64 \nu^2 } + {\mathcal O}(\nu^3)$$$$\begin{array}{lll} &&\frac{1}{\pi}\!\int_0^\pi J_0\Bigl(\kappa \nu^{-1/2} f(\theta^{(n)}){\mathcal G}(t-n\nu)\Bigr)\; d\theta^{(n)}\\ &&=1 - \frac{\kappa^2 f_2\,{\mathcal G}^2(t-n\nu)}{4\nu} + \frac{\kappa^4f_4\,{\mathcal G}^4(t-n\nu)}{64\nu^2} + {\mathcal O}(\nu^3), \end{array}$$$$ \begin{array}{lll}\label{logphi1} \log \Phi_S(\kappa)&=& \log\Bigg( 1 - \frac{\kappa^2 f^2(\theta^{(0)})\,{\mathcal G}^2(t)}{4\nu}\nonumber\\[-6pt]&&{\kern20pt} + \frac{\kappa^4 f^4(\theta^{(0)})\,{\mathcal G}^4(t)}{64 \nu^2 } + {\mathcal O}(\nu^3) \Bigg) \nonumber \\[-6pt] &&+ \sum\limits_{n=-\infty\atop n\neq 0}^\infty \log\Bigg( 1 - \frac{\kappa^2 f_2\,{\mathcal G}^2(t-n\nu)}{4\nu} \nonumber \\[-6pt] && {\kern50pt} + \frac{\kappa^4f_4\,{\mathcal G}^4(t-n\nu)}{64\nu^2} + {\mathcal O}(\nu^3)\Bigg) \nonumber\\ &=& - \frac{\kappa^2 f^2(\theta^{(0)})\,{\mathcal G}^2(t)}{4\nu} - \frac{\kappa^4 f^4(\theta^{(0)})\,{\mathcal G}^4(t)}{64 \nu^2 }\nonumber \\[-6pt] &&- \frac{\kappa^2 f_2}{4\nu} \sum\limits_{n=-\infty\atop n\neq 0}^\infty {\mathcal G}^2(t-n\nu) + \frac{\kappa^4(f_4-2f_2^2)}{64\nu^2} \nonumber\\[-6pt] && \times\sum\limits_{n=-\infty\atop n\neq 0}^\infty {\mathcal G}^4(t\!-\!n\nu) \!+\! {\mathcal O}(\nu^3). \end{array} $$$$\label{scriptg} {\mathcal G}(t)=\nu G_{lgn}(t) + {\mathcal O}(\nu^2). $$$$\label{sumofsquares1} {\mathcal G}_2(t)= \sum\limits_{n=-\infty}^\infty {\mathcal G}^2(t-n\nu) $$$$\label{gensumofsquares} {\mathcal G}_2(t)=2\sum\limits_{n=0}^\infty {\mathcal F}(t+n\nu)\Bigl[{\mathcal F}(t+n\nu)-{\mathcal F}\Bigl(t+(n+1)\nu\Bigr)\Bigr], $$$$\label{gensumofsquares1} {\mathcal G}_2(t)\!=\!-2\nu\sum\limits_{n=0}^\infty {\mathcal F}(t+n\nu)\Bigl[{\mathcal F}'(t\!+\!n\nu)\!+\!\frac{\nu}{2}{\mathcal F}''(t\!+\!n\nu) +\!\frac{\nu^2}{6}{\mathcal F}'''(t\!+\!n\nu)\! +\!\cdots\! \Bigr]. $$$$\label{squares2} {\mathcal G}_2(t)=\nu\int_0^\infty G^2_{lgn}(\xi)\, d\xi +{\mathcal O}(\nu^3).$$$$ \begin{array}{lll}&&{\kern-8pt} -\frac{\kappa^2f_2}{4\nu}\left[{\mathcal G}_2(t)-{\mathcal G}^2(t)\right]\\&&{\kern6pt} =-\frac{\kappa^2 f_2{\mathcal G}_2(t)}{4\nu}+\nu\frac{ \kappa^2 f_2 G^2_{lgn}(t)}{4}+{\mathcal O}(\nu^2)\\[12pt] &&{\kern6pt}=-\frac{\kappa^2 f_2}{4}\int_0^\infty G^2_{lgn}(\xi)\, d\xi +\nu \frac{\kappa^2 f_2 G^2_{lgn}(t)}{4}+{\mathcal O}(\nu^{2})\\[12pt] &&{\kern6pt}= -\frac{\kappa^2}{2}\left(V_0-\nu \frac{ f_2 G^2_{lgn}(t)}{2}\right)+{\mathcal O}(\nu^{2}), \end{array} $$$$ \begin{array}{lll}\label{anothersum} &&{\kern-8pt} -\frac{\kappa^2}{2}\left(V_0+\nu \frac{\left[f^2(\theta^{(0)})- f_2\right] G^2_{lgn}(t)}{2}\right)\nonumber\\&&{\kern6pt} +{\mathcal O}(\nu^{2})= -\frac{\kappa^2 V(t,\theta^{(0)})}{2}+ {\mathcal O}(\nu^{2}), \end{array} $$$$ \begin{array}{lll}\label{yetanothersum} &&{\kern-8pt} \sum\limits_{n=-\infty\atop n\neq 0}^\infty {\mathcal G}^4(t-n\nu)\nonumber\\&&{\kern4pt} =\sum\limits_{n=-\infty}^\infty {\mathcal G}^4(t-n\nu)-{\mathcal G}^4(t)\nonumber\\[3pt]&&{\kern4pt} = \nu^3\int_0^\infty G^4_{lgn}(\xi)\,d\xi + {\mathcal O}(\nu^{4})-\nu^4 G^4_{lgn}(t) + {\mathcal O}(\nu^{5}) \nonumber\\[3pt]&&{\kern4pt} = \nu^3\int_0^\infty G^4_{lgn}(\xi)\,d\xi + {\mathcal O}(\nu^{4}).\end{array}$$$$ \begin{array}{lll}\label{charfunfin} \Phi_S(\kappa)&=&\exp\left[\!-\frac{\kappa^2 V(t,\theta^{(0)})}{2}+\kappa^4 \nu V_1 + {\mathcal O}(\nu^{2})\right]\nonumber\\[12pt]&=&\exp\left[\!-\frac{\kappa^2 V(t,\theta^{(0)})}{2}\right]\left[\!1\!+\! \kappa^4 \nu V_1 \!+\! {\mathcal O}(\nu^{2})\right]\!, \end{array}$$$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty e^{-i \alpha x - \alpha^2/2}\,d\alpha = e^{-x^2/2},$$$$ \begin{array}{lll}\label{fulllgnker} g_{lgn}(t)\!&=&\!\sum\limits_{q=1}^Q g^q_{lgn}(t)\nonumber\\ &=&\sum\limits_{q=1}^Q\Bigg\{R_B\pm\int_{{\mathbb R}^2}dX\,dY \int_{-\infty}^t ds\, \nonumber\\[-4pt] &&{\kern18pt} \times K_{lgn} (|{\bf X}-{\bf X}_q|)G_{lgn}(t-s)\, {\mathcal I}({\bf X},s)\Bigg\}^+, \end{array}$$$$ m(t)\!=\!\left(\frac{V_E}{V_T}-1\right)g_{lgn}(t)\!=\!\left(\frac{V_E}{V_T}-1\right)\sum\limits_{q=1}^Q g^q_{lgn}(t), $$$$\label{firingcomp} M(t,\theta_i)= \left(\frac{V_E}{V_T}-1\right)\sum\limits_{q=1}^Q \left\langle g_{lgn}^q\right\rangle_{\vec\theta,\vec\phi,\theta^{(0)}=\theta_i} (t,\theta_i), $$$$ \begin{array}{lll}\label{glgnq} g_{lgn}^q(t) \!&=&\! \Biggl\{\!R_B \pm I_0\varepsilon\nu^{-\frac{1}{2}}\sum\limits_{n=-\infty}^\infty {\mathcal G}(t-n\nu)\nonumber\\[3pt] &&{\kern2pt} \times\! \int_{{\mathbb R}^2}\!dXdYK_{lgn}(|{\bf X}\!-\!{\bf X}_q|)\sin\!\left[{\bf k}(\theta^{(n)})\cdot{\bf X}\!-\!\phi^{(n)}\right]\!\Biggr\}^+\nonumber\\[3pt] &=\!& \Biggl\{\!R_B \!\pm\! I_0\varepsilon\nu^{-\frac{1}{2}}\!\sum\limits_{n=-\infty}^\infty \!{\mathcal G}(t\!-\!n\nu)\sin\left[{\bf k}(\theta^{(n)})\cdot{\bf X_q}\!-\!\phi^{(n)}\right]\nonumber\\[3pt] &&{\kern2pt} \times \int_{{\mathbb R}^2}dU\,dV\,K_{lgn}(|{\bf Y}|)\cos\left[{\bf k}(\theta^{(n)})\cdot{\bf Y}\right]\Biggr\}^+\nonumber\\[3pt] &=& \Bigg\{R_B \pm I_0\varepsilon\nu^{-\frac{1}{2}}\kappa_{lgn}(k) \sum\limits_{n=-\infty}^\infty {\mathcal G}(t-n\nu)\nonumber\\&&{\kern2pt} \times \sin\left[{\bf k}(\theta^{(n)})\cdot{\bf X_q}-\phi^{(n)}\right]\Bigg\}^+. \end{array} $$$$ \kappa_{lgn}(k)= \int_{{\mathbb R}^2}d\xi\,d\eta\,K_{lgn}\left(\sqrt{\xi^2+\eta^2}\right)\cos k\xi \label{lastint} $$$$ \;\;\; =\int_{{\mathbb R}^2}dU\,dV\,K_{lgn}(|{\bf Y}|)\cos\left[{\bf k}(\theta^{(n)})\cdot{\bf Y}\right] $$$$ \begin{array}{lll}\label{integrateandf} \frac{d}{dt} v^{(i)}_P &=& -g_L \left(v^{(i)}_P - V_{r}\right) -g^{(i)}_{P E}(t)\left(v^{(i)}_P - V_E\right) \nonumber\\&& -g^{(i)}_{PI}(t)\left(v^{(i)}_P - V_I\right), \end{array} $$$$\label{econduct} g^{(i)}_{P E}(t)\! =\! S_{P E} \sum\limits_{j=1}^n K_{PE}\left(\Theta_i\!-\!\Theta_j\right) \sum\limits_k G_E\left(t\! -\! t^j_k\right)\! +\! g_{lgn}^{(i)}(t), $$$$\label{iconduct} g^{(i)}_{P I} (t)\! =\! S_{P I} \sum\limits_{j=1}^n K_{PI}\left(\Theta_i\!-\!\Theta_j\right) \sum\limits_k G_I\left(t\! -\! T^j_k\right) + g_{inh}(t). $$$$ G_P(t)\!=\!\frac{1}{\tau_{Pd}\!-\!\tau_{Pr}}\left[\!\exp\left(\!-\frac{t}{\tau_{Pd}}\right)\!-\!\exp\left(\!-\frac{t}{\tau_{Pr}}\right)\!\right] \:\, \mbox{for} \:\, t\!\geq\! 0, $$$$K_{lgn}(x)=\frac{a}{\pi\sigma_a^2}\exp\left(-\frac{x^2}{\sigma_a^2}\right)-\frac{b}{\pi\sigma_b^2}\exp\left(-\frac{x^2}{\sigma_b^2}\right),$$$$K*F(\theta)=\frac{1}{\pi}\int_{-\pi/2}^{\pi/2} K(\theta-\theta')F(\theta')\,d\theta' .$$$$ \begin{array}{rll}\label{cartoon1} m_P(t,\Theta)&=&f_{lgn}(t,\Theta)+\sum\limits_{P'} C_{PP'}K_{PP'}*G_{P'}*m_{P'} (t,\Theta),\\ P,P'&=&E,I.\end{array} $$$$\label{glgncartoon} f_{lgn}(t,\Theta)= A +B\sum\limits_{n=-\infty}^\infty f(\theta^{(n)}-\Theta){\mathcal G}(t-n\nu). $$$$\label{spatprof1} f(\theta)=\frac{1}{\sqrt{\pi}\sigma_{lgn}}\exp\left(-\frac{\theta^2}{\sigma_{lgn}^2}\right). $$$$ \begin{array}{lll}\label{cartoon2} M_P(t,\theta^{(0)})&=&A+B[f(\theta^{(0)})-\langle f\rangle_\theta]{\mathcal G}(t)\nonumber\\&&+\sum\limits_{P'} C_{PP'}K_{PP'}*G_{P'}*M_{P'}(t,\theta^{(0)}).\end{array} $$$$ \begin{array}{lll}\label{alphabeta} \alpha_P&=&\frac{A(1-C_{PP}+C_{PQ})}{(1-C_{EE})(1-C_{II})-C_{EI}C_{IE}}, \nonumber\\ \beta_P&=&B, \quad P,Q=E,I, \quad P\neq Q, \end{array} $$$$ \begin{array}{rll}\label{linear1} M_P(t,\theta)&=&[f(\theta)-\langle f\rangle_\theta]{\mathcal G}(t)\nonumber\\&&+\sum\limits_{P'} C_{PP'}K_{PP'}*G_{P'}* M_{P'}(t,\theta),\nonumber\\ P,P'&=&E,I, \end{array} $$$$ \label{spatial1} K_{PP'}(\theta)=\frac{1}{\sqrt\pi \sigma_{P'}} \exp\left(-\frac{\theta^2}{\sigma_{P'}^2}\right), $$$$\label{corker1} G_{P}(t)=\left\{\begin{array}{ll}\displaystyle 0, &\quad t\leq 0,\\[6pt] \displaystyle\frac{1}{\tau_{P}}\exp\left(-\frac{t}{\tau_P}\right), &\quad t>0.\end{array}\right. $$$$ \begin{array}{rll} \label{laplaced} \hat M_{P,n}(\lambda)&=&f_n\hat{\mathcal G}(\lambda)+\sum\limits_{P'} C_{PP'} K_{PP',n}\,\hat G_{P'}(\lambda)\,\hat M_{P',n}(\lambda),\nonumber\\ P,P'&=&E,I, \end{array} $$$$ \label{spatialn} K_{PP',n}=\frac{1}{\pi} \exp\left(-\frac{n^2 \sigma_{P'}^2}{4}\right). $$$$\label{lgnkerlap} \hat G_{P}(\lambda)=\frac{1}{\lambda+1/\tau_P}. $$$$\label{hatg} \hat {\mathcal G}(\lambda)=\tau_{lgn}(1-e^{-\nu\lambda})\left(\frac{1}{\lambda+\beta/\tau_{lgn}}-\frac{1}{\lambda+\alpha/\tau_{lgn}}\right). $$$$\label{fn} f_n=\exp\left(-\frac{n^2\sigma_{lgn}^2}{4}\right), \qquad n\neq 0. $$$$ \begin{array}{lll}\label{gensol} &&{\kern-6pt} \hat M_{P,n}(\lambda)\nonumber\\&&=\!\frac{(\lambda\!+\!1/\tau_Q)\left[\!\lambda\!+\!\left(1\!+\!C_{PQ}K_{PQ,n}\!-\!C_{QQ}K_{QQ,n}\right)\!/\tau_P\!\right]}{(\lambda-\lambda_{1,n})(\lambda-\lambda_{2,n})}f_n\hat{\mathcal G}(\lambda),\nonumber\\ &&{\kern-6pt} Q\neq P \end{array} $$$$ \lambda_{1,n}, \, \lambda_{2,n}=\frac{\lambda_{E,n}+\lambda_{I,n}}{2} \pm\sqrt{\frac{\left(\lambda_{E,n}-\lambda_{I,n}\right)^2}{4} +\frac{C_{IE}K_{IE,n}C_{EI}K_{EI,n}}{\tau_E\tau_I}} $$$$\label{vertex} \lambda_{P,n}=\frac{C_{PP}K_{PP,n}-1}{\tau_P}, \qquad P=E,I. $$$$ \begin{array}{lll} \hat \Gamma_{\!P,n}(\lambda)&\!=\!\!&f_n\tau_{lgn}\frac{(\lambda\!+\!1/\tau_Q)\left[\!\lambda\!+\!\left(1\!+\!C_{PQ}K_{PQ,n}\!-\!C_{QQ}K_{QQ,n}\right)\!/\tau_P\!\right]}{(\lambda-\lambda_{1,n})(\lambda-\lambda_{2,n})}\nonumber\\[6pt] &&\times \left(\frac{1}{\lambda+\beta/\tau_{lgn}}-\frac{1}{\lambda+\alpha/\tau_{lgn}}\right) \end{array} $$$$\label{mpndiff} M_{P,n}(t)=\Gamma_{P,n}(t)-\Gamma_{P,n}(t-\nu), $$$$ \Gamma_{P,n}(t)= \left\{ \begin{array}{ll} 0, & t<0, \\[6pt] \mbox{sum of the residues of } \hat \Gamma_{P,n}(\lambda)e^{\lambda t},\quad & t\geq 0. \end{array} \right. $$$$ \begin{array}{lll}\label{gammapn} &&\!\!\! \Gamma_{P,n}(t)=\frac{f_n\tau_{lgn}}{\tau_E\tau_I}\left\{\sum\limits_{{\mathop {j = 1}\limits_{i \ne j} }}^2\frac{\tau_{lgn}(\alpha-\beta)(1+\tau_P\lambda_j)\left(1+C_{PQ}K_{PQ,n}-C_{QQ}K_{QQ,n}+\tau_Q\lambda_j\right)}{(\lambda_j-\lambda_i)(\alpha+\tau_{lgn}\lambda_j)(\beta+\tau_{lgn}\lambda_j)}e^{\lambda_j t} \right.\nonumber\\[6pt] &&\frac{(\tau_{lgn}-\tau_P\alpha)\left[\tau_{lgn} (1+C_{PQ}K_{PQ,n}-C_{QQ}K_{QQ,n})-\tau_Q\alpha\right]}{(\alpha+\lambda_1\tau_{lgn})(\alpha+\lambda_2\tau_{lgn})}e^{-\alpha t/\tau_{lgn}}\nonumber \\[6pt] &&\left.+\frac{(\tau_{lgn}-\tau_P\beta)\left[\tau_{lgn} (1+C_{PQ}K_{PQ,n}-C_{QQ}K_{QQ,n})-\tau_Q\beta\right]}{(\beta+\lambda_1\tau_{lgn})(\beta+\lambda_2\tau_{lgn})}e^{-\beta t/\tau_{lgn}}\!\right\}, \end{array} $$$$\label{mfourier} M_P(t,\theta)=\sum_{n=-\infty}^\infty M_{P,n}(t) e^{2in\theta}. $$$$\label{timefactors} \Lambda(t,\gamma,\nu)=\left\{\begin{array}{ll} 0, & {\kern15pt} t\ \ \ \ <-\nu,\\ (e^{\gamma(t+\nu)}-1)/\gamma,\qquad &{\kern2pt} -\nu\leq t< 0,\\ e^{\gamma t }(e^{\gamma \nu}-1)/\gamma, & {\kern15pt}t\ \ \ \ \geq 0. \end{array}\right. $$

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