A minimal mechanistic model for temporal signal processing in the lateral geniculate nucleus

Abstract

The receptive fields of cells in the lateral geniculate nucleus (LGN) are shaped by their diverse set of impinging inputs: feedforward synaptic inputs stemming from retina, and feedback inputs stemming from the visual cortex and the thalamic reticular nucleus. To probe the possible roles of these feedforward and feedback inputs in shaping the temporal receptive-field structure of LGN relay cells, we here present and investigate a minimal mechanistic firing-rate model tailored to elucidate their disparate features. The model for LGN relay ON cells includes feedforward excitation and inhibition (via interneurons) from retinal ON cells and excitatory and inhibitory (via thalamic reticular nucleus cells and interneurons) feedback from cortical ON and OFF cells. From a general firing-rate model formulated in terms of Volterra integral equations, we derive a single delay differential equation with absolute delay governing the dynamics of the system. A freely available and easy-to-use GUI-based MATLAB version of this minimal mechanistic LGN circuit model is provided. We particularly investigate the LGN relay-cell impulse response and find through thorough explorations of the model’s parameter space that both purely feedforward models and feedback models with feedforward excitation only, can account quantitatively for previously reported experimental results. We find, however, that the purely feedforward model predicts two impulse response measures, the time to first peak and the biphasic index (measuring the relative weight of the rebound phase) to be anticorrelated. In contrast, the models with feedback predict different correlations between these two measures. This suggests an experimental test assessing the relative importance of feedforward and feedback connections in shaping the impulse response of LGN relay cells.

Appendices

Appendix 1: Derivation of model on differential form

In the present appendix we derive the differential Eq. (17) for the dynamical variable z(t) defined in (16) when the coupling kernels h ^ON/OFF_cr and h ^ON/OFFx_rc in (12) and (13) are assumed.

Insertion of (12) into (7) gives, by means of (6),

$$ \begin{aligned} R_{\rm c}^{\rm ON}(t)&= F_{\rm c}^{\rm ON}\left(w_{\rm cr}^{\rm ON}\int\limits_{-\infty}^{t}h_{\rm cr}^{\rm ON}(t-s) \hat{R}_{\rm r}^{\rm ON}(s)ds \right)\\ &=F_{\rm c}^{\rm ON}\left(w_{\rm cr}^{\rm ON}\int\limits_{-\infty}^{t}\delta((t-\Updelta_{\rm cr})-s) \hat{R}_{\rm r}^{\rm ON}(s)ds \right)\\ &= F_{\rm c}^{\rm ON}\left(w_{\rm cr}^{\rm ON}\hat{R}_{\rm r}^{\rm ON}(t-\Updelta_{\rm cr})\right)\\ &=F_{\rm c}^{\rm ON}\left(w_{\rm cr}^{\rm ON} F_{\rm r}^{\rm ON}\left(A_{\rm g} w_{\rm rg}^{\rm ON}\left(\bar{i}_{\rm r0}^{\rm ON}+\bar{r}_{\rm g}(t-\Updelta_{\rm cr})+z(t-\Updelta_{\rm cr}) \right)\right)\right) \end{aligned} $$

(56)

where we also have used the definition for $\bar{r}_{\rm g}(t)$ in (18) and $\bar{i}_{{\rm r0}}^{\rm ON}$ in (21). (Note that the notation $\hat{R}_{\rm c}^{\rm ON}(t)$ in (7) has been replaced by R ^ON_c (t) in (56) since the background firing rates of the cortical cells have been assumed zero.) Correspondingly, we find that

$$ R_{\rm c}^{\rm OFF}(t) = F_{\rm c}^{\rm OFF}\left(w_{\rm cr}^{\rm OFF}F_{\rm r}^{\rm OFF} \left(A_{\rm g} w_{\rm rg}^{\rm ON}\left(\bar{i}_{\rm r0}^{\rm OFF}-\bar{r}_{\rm g}(t-\Updelta_{\rm cr})-z(t-\Updelta_{\rm cr})\right)\right)\right), $$

(57)

and we have arrived at (25) and (26), respectively.

Differentiation of the expression for the differential delay equation with absolute delay z in (16) with respect to t gives

$$ \begin{aligned} \frac{d}{dt}z(t)&=\frac{w_{\rm rc}^{\rm ON}}{A_{\rm g} w_{\rm rg}^{\rm ON}} \frac{d}{dt} \left((h_{\rm rc}^{\rm ON}\ast R_{\rm c}^{\rm ON})(t)\right)+\frac{w_{\rm rc}^{\rm OFFx}}{A_{\rm g} w_{\rm rg}^{\rm ON}} \frac{d}{dt}\left((h_{\rm rc}^{\rm OFFx}\ast R_{\rm c}^{\rm OFF})(t)\right)\\ &=\frac{w_{\rm rc}^{\rm ON}}{A_{\rm g} w_{\rm rg}^{\rm ON}} \frac{d}{dt}\left(\int\limits_{-\infty}^t \frac{e^{-(t-\Updelta_{\rm rc}-s)/\tau_{\rm rc}}}{\tau_{\rm rc}} \Uptheta(t-\Updelta_{\rm rc}-s) R_{\rm c}^{\rm ON}(s)ds\right)\\ &\quad +\frac{w_{\rm rc}^{\rm OFFx}}{A_{\rm g} w_{\rm rg}^{\rm ON}} \frac{d}{dt}\left(\int\limits_{-\infty}^t \frac{e^{-(t-\Updelta_{\rm rc}-s)/\tau_{\rm rc}}}{\tau_{\rm rc}} \Uptheta(t-\Updelta_{\rm rc}-s) R_{\rm c}^{\rm OFF}(s)ds \right) \end{aligned} $$

(58)

where we have used (13). Since $\Uptheta(t-\Updelta_{\rm rc}-s)=1$ only for $s< t-\Updelta_{\rm rc}$, the integrals in (58) can be rewritten as

$$ \int\limits_{-\infty}^{t}\frac{e^{-(t-\Updelta_{\rm rc}-s)/\tau_{\rm rc}}} {\tau_{\rm rc}} \Uptheta(t-\Updelta_{\rm rc}-s)R_{\rm c}^{\rm ON/OFF}(s)ds=\int\limits_{-\infty}^u \frac{e^{-(u-s)/\tau_{\rm rc}}} {\tau_{\rm rc}} R_{\rm c}^{\rm ON/OFF}(s)ds, $$

(59)

where we have introduced $u=t-\Updelta_{\rm rc}$. By means of the general differentiation rule

$$ \frac{d}{dt}\left(\int\limits^{t}_{a}F(t,s)ds\right)= F(t,t) + \int\limits^{t}_{a} \frac{\partial}{{\partial}t} F(t,s)ds $$

(60)

we now find for the first integral in (58),

$$ \begin{aligned} &\frac{w_{\rm rc}^{\rm ON}}{A_{\rm g} w_{\rm rg}^{\rm ON}} \frac{d}{du}\left[\int\limits_{-\infty}^u \frac{e^{-(u-s)/\tau_{\rm rc}}}{\tau_{\rm rc}} R_{\rm c}^{\rm ON}(s)ds \right]\\ &\quad =\frac{w_{\rm rc}^{\rm ON}}{A_{\rm g} w_{\rm rg}^{\rm ON}} \frac{1}{\tau_{\rm rc}}\left(R_{\rm c}^{\rm ON}(u)-\int\limits_{-\infty}^u \frac{e^{-(u-s)/\tau_{\rm rc}}}{\tau_{\rm rc}} R_{\rm c}^{\rm ON}(s)ds \right)\\ &\quad=\frac{w_{\rm rc}^{\rm ON}}{A_{\rm g} w_{\rm rg}^{\rm ON}} \frac{1}{\tau_{\rm rc}} \left(R_{\rm c}^{\rm ON}(t-\Updelta_{\rm rc}) -\int\limits_{-\infty}^{t-\Updelta_{\rm rc}} \frac{e^{-(t-\Updelta_{\rm rc}-s)/\tau_{\rm rc}}}{\tau_{\rm rc}} R_{\rm c}^{{\rm ON}}(s)ds \right)\\ &\quad=\frac{w_{\rm rc}^{\rm ON}}{A_{\rm g} w_{\rm rg}^{\rm ON}}\frac{1}{\tau_{\rm rc}} \left(R_{\rm c}^{\rm ON}(t-\Updelta_{\rm rc})-\int\limits_{-\infty}^{t}\frac{e^{-(t-\Updelta_{\rm rc}-s)/\tau_{\rm rc}}}{\tau_{\rm rc}}\Uptheta(t-\Updelta_{\rm rc}-s) R_{\rm c}^{\rm ON}(s)ds \right)\\ &\quad =\frac{w_{\rm rc}^{\rm ON}}{A_{\rm g} w_{\rm rg}^{\rm ON}} \frac{1}{\tau_{\rm rc}}\left(R_{\rm c}^{{\rm ON}}(t-\Updelta_{\rm rc})-(h_{\rm rc}^{\rm ON} \ast R_{\rm c}^{\rm ON})(t)\right) \end{aligned} $$

(61)

For the OFF integral in (58) we correspondingly find

$$ \frac{w_{\rm rc}^{\rm OFFx}}{A_{\rm g} w_{\rm rg}^{\rm ON}} \frac{d}{du}\left[\int\limits_{-\infty}^u \frac{e^{-(u-s)/\tau_{\rm rc}}}{\tau_{\rm rc}} R_{\rm c}^{\rm OFF}(s)ds \right]= \frac{w_{\rm rc}^{\rm OFFx}}{A_{\rm g} w_{\rm rg}^{\rm ON}} \frac{1}{\tau_{\rm rc}}\left(R_{\rm c}^{\rm OFF}(t-\Updelta_{\rm rc})-(h_{\rm rc}^{\rm OFFx} \ast R_{\rm c}^{\rm OFF})(t)\right) $$

(62)

With the use of the definition for z(t) in (16) we thus find that the differential equation for z(t) in (58) simplifies to

$$ \frac{d}{dt} z(t)=-\frac{z(t)}{\tau_{\rm rc}}+\frac{w_{\rm rc}^{\rm ON}}{A_{\rm g} w_{\rm rg}^{\rm ON}\tau_{\rm rc}} R_{\rm c}^{\rm ON}(t-\Updelta_{\rm rc})+\frac{w_{\rm rc}^{\rm OFFx}}{A_{\rm g} w_{\rm rg}^{\rm ON}\tau_{\rm rc}} R_{\rm c}^{\rm OFF}(t-\Updelta_{\rm rc}) $$

(63)

Use of the expressions for R ^ON_c (t) and R ^OFF_c (t) in (56) and (57), respectively, and the definition $\Updelta_{\rm fb} \equiv \Updelta_{\rm rc}+\Updelta_{\rm cr}$ in (20), then gives the delay differential equation with absolute delay (17) for z(t) in the main text.

Appendix 2 : Threshold firing-rate functions

Use of threshold firing-rate functions as described in (27)–(30) in the expression for R ^ON_c (t) in (25) gives

$$ R_{\rm c}^{\rm ON}(t)=A_{\rm g} w_{\rm rg}^{\rm ON} w_{\rm rg}^{\rm ON}\left[\left[\bar{i}_{\rm r0}^{\rm ON}+\bar{r}_{\rm g}(t-\Updelta_{\rm cr})+ z(t-\Updelta_{\rm cr})-\bar{\lambda}_{\rm r}^{\rm ON}\right]_+ -\bar{\lambda}_{\rm c}^{\rm ON}\right]_+ $$

(64)

where we have introduced $\bar{\lambda}_{\rm r}^{\rm ON} \equiv \Uplambda_{\rm r}^{\rm ON}/(A_{\rm g} w_{\rm rg}^{\rm ON})$ and $\bar{\lambda}_{\rm c}^{\rm ON} \equiv \Uplambda_{\rm c}^{\rm ON}/(A_{\rm g} w_{\rm rg}^{\rm ON} w_{\rm cr}^{\rm ON})$. The requirement of zero background firing rate for the cortical ON cells translates to the constraint

$$ \bar{\lambda}_{\rm c}^{\rm ON} \geq \left[\bar{i}_{\rm r}^{\rm ON} - \bar{\lambda}_{\rm r}^{\rm ON}\right]_+ =\left[-\lambda_{\rm r}^{\rm ON}\right]_+ \geq 0, $$

(65)

where we have used the definition of λ ^ON_r in (39).

Since we enforce that $\bar{\lambda}_{\rm c}^{\rm ON} \geq 0$, the innermost half-wave rectification in (64) can be removed. (For the cases where $\left(\bar{r}_{\rm g}(t-\Updelta_{\rm cr})+ z(t-\Updelta_{\rm cr})-\lambda_{\rm r}^{\rm ON}\right) < 0$, the function (64) will be zero also with the innermost half-wave rectification function replaced by the linear function). The expression (64) thus simplifies to

$$ R_{\rm c}^{\rm ON}(t)=A_{\rm g} w_{\rm rg}^{\rm ON} w_{\rm cr}^{\rm ON} \left[\bar{r}_{\rm g}(t-\Updelta_{\rm cr})+ z(t-\Updelta_{\rm cr})- \lambda_{\rm c}^{{\rm ON}}\right]_+ $$

(66)

corresponding to (37) in the main text. Here the definition of λ ^ON_c0 in (33) has been used. A corresponding argument for R ^OFF_c (t) gives

$$ R_{\rm c}^{\rm OFF}(t)=A_{\rm g} w_{\rm rg}^{\rm ON} w_{\rm cr}^{\rm OFF}\left[-\left(\bar{r}_{\rm g}(t-\Updelta_{\rm cr})+ z(t-\Updelta_{\rm cr})-\lambda_{\rm c}^{\rm OFF}\right)\right]_+ $$

(67)

corresponding to (38) in the main text. Insertion of these new expression for R ^ON_c (t) and R ^OFF_c (t) into

$$ \frac{d}{dt}z(t)=-\frac{z(t)}{\tau_{\rm rc}}+\frac{w_{\rm rc}^{\rm ON}}{A_{\rm g} w_{\rm rg}^{\rm ON}\tau_{\rm rc}} R_{\rm c}^{\rm ON}(t-\Updelta_{\rm rc})+\frac{w_{\rm rc}^{\rm OFFx}}{A_{\rm g} w_{\rm rg}^{\rm ON}\tau_{\rm rc}} R_{\rm c}^{\rm OFF}(t-\Updelta_{\rm rc}) $$

(68)

then gives (31) in the main text.

Appendix 3: Numerical methods

General: In the numerical investigations we needed to solve the delay differential equation in (31). This was done in MATLAB using the routine dde23 (and in the case of zero delay the routine ode45 ). The auxiliary variable z(t) was set to be zero for all times prior to stimulus onset.

Simulation of drifting-grating responses: In numerical evaluations of the responses to drifting gratings, the transient part of the response following stimulus onset was omitted. The MATLAB routine fft implementing the fast Fourier transform was then used to extract the first harmonic component from the steady-state part of the response.

Simulations of impulse response: We first simulated the first 250 ms of impulse response after onset of the retinal input, cf. Fig. 4 to obtain t _max and I _BP. If four or more phases were present in these first 250 ms, we evaluated the impulse-response for 250 ms more. The normalized rebound magnitude (NRM) was then found by evaluating the integrals over the first and all subsequent phases, respectively, using the MATLAB routine quadl . However, if the area of the fourth phase was found to be larger than one hundredth of the area of the first phase, NRM was not calculated, cf. white regions in the phase plots in Figs. 7 and 8.