Conductance-based refractory density model of primary visual cortex

Abstract

A layered continual population model of primary visual cortex has been constructed, which reproduces a set of experimental data, including postsynaptic responses of single neurons on extracellular electric stimulation and spatially distributed activity patterns in response to visual stimulation. In the model, synaptically interacting excitatory and inhibitory neuronal populations are described by a conductance-based refractory density approach. Populations of two-compartment excitatory and inhibitory neurons in cortical layers 2/3 and 4 are distributed in the 2-d cortical space and connected by AMPA, NMDA and GABA type synapses. The external connections are pinwheel-like, according to the orientation of a stimulus. Intracortical connections are isotropic local and patchy between neurons with similar orientations. The model proposes better temporal resolution and more detailed elaboration than conventional mean-field models. In comparison to large network simulations, it excludes a posteriori statistical data manipulation and provides better computational efficiency and minimal parametrization.

Appendices

Appendix A: Single population CBRD model response to a current-step stimulation

The most important advantage of the CBRD model against the firing rate models is the accuracy of transient solutions. We demonstrate this property in a simple case of a single population of single-compartment leaky integrate-and-fire (LIF) neurons stimulated by a stepwise input current (Fig. 13). As seen from the figure, the CBRD model (red solid line) well approximates the solution obtained by Monte-Carlo simulation with 4000 individual neurons (or trials) (thin fluctuating line). In contrary, the simplest firing-rate models fail to reproduce the beginning and the end of the Monte-Carlo solution. It is so, because the firing-rate models implicitly assume that the neurons within a population are always de-synchronized. This assumption is wrong, for instance, at the beginning of stimulation, when almost all neurons respond by their first spikes rather synchronously, that creates a bump in the firing rate response seen in Fig. 13. All the responses converge to the same steady-state solution derived by Johannesma (1968), including the CBRD model which does not explicitly use that solution. The precision of the CBRD model in the steady states is less than 1 Hz for the whole ranges of input current and conductance (data not shown). As to the costs of calculations, the CBRD method require about 100 points of discretization in $t^{*}$-space, thus it is computationally equivalent to that number of neurons, whereas the Monte-Carlo simulations require many thousands of neurons to get a convergent solution with 1 ms precision. Thus the CBRD approach is in $10\div 100$ times more efficient than the direct simulations.

Fig. 13

The firing rate of a population of the stochastic leaky integrate-and-fire (LIF) neurons as a response to the current step stimulation. The firing rate was calculated by 4 models: Monte-Carlo simulation of 4000 individual trials, the firing-rate (FR) model based on equation $\nu (t)=\nu ^{ss}(I(t))$, where $\nu ^{ss}$ is the steady state solution for a LIF with noise; the FR model based on the relaxation equation $\tau _{m} ~d\nu /dt=\nu ^{ss}(I)-\nu$, and the CBRD model based on Eqs. (1, 2). The parameters were: $C=0.1~nF$, ${V_{th}}-V_{rest}=10$ mV, $V_{reset}=V_{rest}$, $\tau _{m}=10$ ms, $I(t)=\{1.5~\mu$A/cm$^{2}$ for t from 0 to 100 ms, and 0 otherwise$\}$, $\sigma _V=2$ mV

Similar test problems with the comparison to Monte-Carlo simulations were solved for the populations of fast spiking and regular spiking neurons in Chizhov and Graham (2007, 2008).

Appendix B: Effect of population model type on simulation of a postsynaptic response

In order to clarify the importance of CBRD model application, let’s compare a CBRD-based model with a firing-rate (FR) model in the case of a recurrent network consisting of two populations of excitatory and inhibitory LIF neurons receiving an external stimulus. All neurons are the same, with the parameters given in the caption to Fig. 13. The both populations receive AMPA-mediated input $\left (\overline {g}_{ampa,Th,E}=\overline {g}_{ampa,Th,I}=0.6 ~mS/cm^{2}\right )$. The excitatory population receives also the inhibitory recurrent input with $\overline {g}_{gaba,I,E}=0.6 ~mS/cm^{2}$. The synaptic kinetics is given by the parameters in Section 2.9. The stimulus was set by the rate $\varphi _{Th, E}=100$ Hz during 1 ms. The FR model was taken in the form $\nu =\nu ^{ss}(U)$ with the voltage U governed by the equation of LIF model without a renewal at threshold, and $\nu ^{ss}$ is the steady state solution (Johannesma 1968).

The stimulation evokes an EPSP in the inhibitory population (middle plot in Fig. 14) and a compound PSP in the excitatory population (top plot). The PSP in inhibitory cells determines the firing rate $\nu _{I}$ (bottom plot), which, in its turn, determines inhibitory component of the PSP in the excitatory population. The results obtained in the CBRD- and FR models are different. The firing rate in the FR model follows the voltage trace. The rate is underestimated at the stage of depolarization. That is why, the inhibitory component of the PSP in the inhibitory neurons is delayed and smaller in the FR-model than in the CBRD-model. The FR model can be, however, improved by using it modified version proposed in Chizhov (2002) and Buchin et al. (2010). The modified model well describes the synchronized burst caused by initial spikes and it wrong for the second part of the response, because it can not correctly reflect the effects of neuronal refractoriness.

Fig. 14

Postsynaptic potentials (PSPs) and firing rate in the network of two, excitatory and inhibitory, LIF-neuron populations, calculated by the CBRD and firing-rate models

Appendix C: Postsynaptic potentials in two-compartment model

The 2-compartment pyramidal neuron model from Chizhov (2004) relates a somatic synaptic current registered in the conditions of fixed voltage on soma (voltage-clamp) to a somatic voltage registered in the current-clamp conditions. These two regimes, corresponding to two different boundary problems for any spatially distributed neuron, are described by the cable equation. The 2-compartment model equations are derived as an approximation of the equations of the two joint boundary problems for a passive cylindrical dendrite and a concentrated passive soma. The approximation assumes a linear distribution of the voltage along the dendrite with the extreme values of U at the soma and $U_{d}$ at the distal end of the dendrite. The first boundary problem corresponds to the case of somatic voltage-clamp regime of the synaptic current ($I_{VC}$) measurement when the synaptic current at the distal part of the dendrite ($I_{dendr}$) is unknown. The solution of the reverse problem for $I_{dendr}$ is expressed by the differential equation, $I_{Dendr}=({\tau _{m}^{0}}/{2} ~{dI_{VC}}/{dt} +{3I_{VC}}/{2}$. The estimate of $I_{dendr}$ is then substituted in the equations of the second boundary problem corresponding to the current-clamp regime when the somatic voltage is determined by the synaptic current at the distal end of the dendrite, $I_{dendr}$. The precision of $I_{dendr}$ estimation by the 2-compartment model is rough in comparison with the original synaptic current in the distributed model, nevertheless due to the joint solution of the direct and reverse problems the approximation of the somatic voltage U, given the somatic current $I_{VC}$, is much more precise.

Here we write down the two-compartment model with only two synaptic sources, the AMPA-mediated channels attributed to the dendrite and the GABA-mediated channels at soma. The two-compartment linearized model is described by the reduced version of Eqs. (2, 3), i.e.

$$\begin{array}{@{}rcl@{}} C\frac{dU}{dt} &=& -g_{L}(U-V_{rest}) + \gamma_{d} ~g_{L}(U_{d}-U) \\ &&+ g_{gaba}(t)(V_{gaba}-U^{E}) +I^{el.} \end{array} $$

(43)

$$\begin{array}{@{}rcl@{}} C\frac{dU_{d}}{dt}&=&-g_{L}(U_{d}-V_{rest})-(2+\gamma_{d})g_{L}(U_{d}-U) \\ &&+ \frac{1}{\gamma_{d}} \left({\tau_{m}^{0}} \frac{d}{dt} +{3}\right)\left( g_{ampa}(t)(V_{ampa}-U) \right),\\ \end{array} $$

(44)

The effect of taking into account the second compartment on approximation of postsynaptic potentials is shown in Fig. 15. One- and two-compartment passive neurons were stimulated by a short electric pulse which excites the presynaptic excitatory and inhibitory fibers.

Fig. 15

AMPA- and GABA-mediated and compound postsynaptic somatic currents (PSCs) and potentials (PSPs), calculated by one- and two-compartment passive models as responses to short pulse stimulation of presynaptic fibers. The neuron passive parameters and synaptic conductance kinetics correspond to the ones of the excitatory neurons in the full CBRD-model. The maximum synaptic conductances were $\overline {g}_{ampa}=0.6 ~mS/cm^{2}$ and $\overline {g}_{ampa}=1.2 ~mS/cm^{2}$, except of the case with the decreased $\overline {g}_{ampa}=0.35 ~mS/cm^{2}$, marked by the gray dashed line. Also, PSCs are the same in one- and two-compartment models, except of the one with the decreased $\overline {g}_{ampa}$

As seen from Fig. 15, the amplitudes of PSPs are different according to different input conductances. For one-compartment model the input conductance is equal to $C/\tau _{m}^{0}$, whereas for the somatic or dendritic inputs in two-compartment neuron it is either $(3+2~\gamma _{d})/(3+\gamma _{d})~C/\tau _{m}^{0}$ or $(1+2/3~\gamma _{d})~C/\tau _{m}^{0}$, correspondingly. The temporal characteristics are also slightly different, and it is crucial for the summation of excitatory and inhibitory components. Comparison of the compound PSPs reveals the difference of the responses of 1-compartment and 2- compartment neurons (dashed and solid black lines). Moreover, even if the maximum AMPA-conductance in 1-compartment model fits the maximum of the response of the 2-compartment model (gray dashed line), the transient behavior of the 1-compartment model is delayed. The voltage rate, that affects the firing rate and consequent recurrent activity, might significantly differ for the two models. Thus, the implementation of the 2-compartment model into the CBRD-model is important for its fitting to experiments.

Appendix D: Model of extracellular stimulation

Stimulation by an extracellular electrode is often used to study evoked neuronal activity. Here we are constructing a simple model describing the dependence of a number of excited spiking neurons on the stimulation amplitude and distance to the electrode. A typical stimulating current pulse is characterized by the amplitude $I^{ext}$ of the order of 10$\mu$A and a duration $\Delta t^{ext}$ of the order of 0.1ms. We assume that the cortical tissue is an infinite homogeneous conductive medium, where a point current source is located at $(x_{el},y_{el})$. A current pulse creates an electric field with an extracellular potential distribution reverse proportional to the distance to the electrode (Rattay 1999), $r=\sqrt {(x-x_{el})^{2}+(y-y_{el})^{2}}$, its amplitude being proportional to the stimulating current $I^{ext}$ with a coefficient $k_{1}^{ext}$:

$$V_{e}(r)=\frac{k_{1}^{ext} I^{ext}}{r}. $$

(45)

We assume that this potential is constant during the pulse and equal to zero after the stimulus. A gradient of the extracellular potential creates an instantaneous change of the transmembrane potential in every neuron, $\Delta V(r)$. We assume $\Delta V(r)$ to be proportional to the electric field strength (Gimsa et al. 2006), which is proportional to the gradient of $V_{e}(r)$, i.e. $\Delta V(r)\sim grad~V_{e}(r)$. Substituting Eq. (45), we get

$$\Delta V(r)=\frac{k^{ext} I^{ext}}{r^{2}}$$

(46)

with a new coefficient $k^{ext}$.

According to the threshold model of a single neuron, a spike occurs when the membrane potential reaches the threshold ${V_{th}}$. Assuming that initially all neurons are in the resting state, and measuring ${V_{th}}$ relative to the rest potential, we deduce that a spike in a neuron located at r occurs if its transmembrane potential perturbation $\Delta V^{\prime }(r)$ is greater than the threshold ${V_{th}}$. Assuming the distribution of $\Delta V^{\prime }(r)$ to be gaussian with the mean $\Delta V(r)$ and the dispersion $\sigma _{ext}$, the fraction of firing neurons can be estimated as

$$\begin{array}{@{}rcl@{}} N(r)&=&\left( 1- \int_{-\infty}^{T_{ext}(r)} e^{-u^{2}} ~du \left/ \int_{-\infty}^{T_{0}} e^{-u^{2}} ~du \right.\right) \\ &=&\frac{\text{erf}(T_{0})-\text{erf}(T_{ext})}{1+\text{erf}(T_{0})}, \end{array} $$

(47)

where $u=\Delta V'(r)/\sqrt {2}\sigma _{ext}$, $~T_{0}={V_{th}}/\sqrt {2}\sigma _{ext}$, $T_{ext}(r)=({V_{th}}-\Delta V(r))/\sqrt {2}\sigma _{ext}$. Assuming that the spike time moments are uniformly distributed across the interval [0, $\Delta t^{ext}$] and substituting Eq. (46) into Eq. (47), we obtain the estimation for the firing rate

$$\begin{array}{@{}rcl@{}} \nu^{ext}(t,r)&=&\left\{ \frac{1}{\Delta t^{ext}} \frac{\text{erf}{(T_{0})}+\text{erf}{(S^{ext}/r^{2}-T_{0})}}{1+\text{erf}{(T_{0}})},\right. \\ &&\left.\vphantom{\frac{\text{erf}{(T_{0})}+\text{erf}{(S^{ext}/r^{2}-T_{0})}}{1+\text{erf}{(T_{0}})}} \text{if}~~~0<t<\Delta t^{ext}; ~~~0, ~~~\text{otherwise} \right\}, \end{array} $$

(48)

where $S^{ext}=k^{ext} I^{ext}/(\sqrt {2} \sigma _{ext})$ means an effective area of stimulation and plays a role of a specific amplitude of stimulation. Here $I^{ext}$ and $\Delta t^{ext}$ are known from experimental settings; ${V_{th}}$ is estimated as the characteristic spike threshold potential of neurons; the coefficients $\sigma _{ext}$, $k^{ext}$ are to be matched. Thus, the Eq. 48 has only 3 parameters of stimulation, the duration $\Delta t^{ext}$, the amplitude $S^{ext}$ and the threshold $T_{0}$. Qualitatively, the dependence Eq. (48) is shown in Fig. 16. For instance, we set the threshold $T_{0}=1$ and fix $S^{ext}$. The profile has a plateau for small r, where $\nu ^{ext}(t,r) ~\Delta t^{ext}$ is about 1, meaning that all neurons located near the electrode give spikes. The fraction of spiking neurons, $\nu ^{ext}(t,r) ~\Delta t^{ext}$ decreases with distance, thus more remote neurons are not activated. Concluding, we note that the obtained formula is qualitatively consistent with the profile of excitation evoked by extracellular electrode. The formula provides minimum number of the parameters governing the effect, and the simulations made in the Results section do not reveal dramatic sensitivity to values of the parameters of Eq. (48).

Fig. 16

Fraction of extracellularly excited neurons as a function of the distance to the stimulating electrode, r, related to the effective area of stimulation, $S^{ext}$, for different threshold value $T_{0}$

Appendix E: Matrix of synaptic connections

In principle, estimates of maximum synaptic conductances can be obtained from experimental data of intracellular registrations in pairs of neurons, given, for example, by Buhl et al. (1997), Yoshimura and Callaway (2005), Bannister and Thomson (2006), Thomson et al. (2002), Thomson and Lamy (2007), Lubke and Feldmeyer (2007), however not all necessary parameters are present in those papers. Here we derive a formula for the estimation and consider its dependence on parameters.

The maximum synaptic conductance $\overline {g}_{S,i,j}$ is determined by the following parameters: $P^{hit}_{i,j}$, the hit rate (Thomson and Lamy 2007), i.e. the fraction of connected versus unconnected pairs of i-type presynaptic and j-type postsynaptic neurons; $\rho _{i}$, the density of i-type neurons in a unit area of the cortical surface; $V^{PSP}_{i,j}$, the mean postsynaptic potential measured in j-type neurons connected to i-type presynaptic neurons; $a_{i,j}$, the radius of the spot where the pairs were searched; $d_{i,j}$, the characteristic length of $(i,j)$-type local isotropic connections as defined by Eq. (38).

The maximum synaptic conductance $\overline {g}_{S,i,j}$ is proportional to the mean conductance of such single connection, $g^{1}_{i,j}$, and the number of connected $(i,j)$-type pairs, $N^{conn}_{i,j}$, i.e.

$$\begin{array}{@{}rcl@{}} {\overline{g}_{S,i,j}}=N^{conn}_{i,j} g^{1}_{i,j}, \end{array} $$

(49)

where $N^{conn}_{i,j}$ is proportional to the integral over the cortical space of the connection probability of j-type postsynaptic cell with i-type presynaptic one, $P^{conn}_{i,j}(r)$, which depends on the distance between the cells r, i.e.

$$\begin{array}{@{}rcl@{}} N^{conn}_{i,j} = 2\pi \int_{0}^{\infty} P^{conn}_{i,j}(r) \rho_{i} r dr. \end{array} $$

(50)

According to Eq. (38), the profile of connection probability is

$$P^{conn}_{i,j}(r)=P^{conn}_{i,j}(0) ~\exp\left(-\left(r/d_{i,j}\right)^{2}\right). $$

(51)

The coefficient $P^{conn}_{i,j}(0)$ can be estimated from the hit rate which is

$$\begin{array}{@{}rcl@{}} P^{hit}_{i,j}&=&\left(2 \pi \int_{0}^{a_{i,j}} P^{conn}_{i,j}(r) ~r ~dr \right)/\left(\pi a_{i,j}^{2}\right) \\ &=&\frac{2}{a_{i,j}^{2}} ~P^{conn}_{i,j}(0) \int_{0}^{a_{i,j}} ~\exp\left(-\left(r/d_{i,j}\right)^{2}\right)~ r ~dr \\ &=&P^{conn}_{i,j}(0) ~(1-\exp \left(-\left(a_{i,j}/d_{i,j}\right)^{2}\right). \end{array} $$

(52)

Now we rewrite Eq. (51) as

$$P^{conn}_{i,j}(r){}={}\frac{P^{hit}_{i,j}}1{} -{}\exp {}\left({}-{}\left(a_{i,j}/d_{i,j}\right)^{2}\right) \exp{}\left({}-{}\left(r/d_{i,j}\right)^{2}\right). $$

(53)

Postsynaptic potentials are recorded in current clamp mode. Let’s assume that in this case the synaptic current is balanced with the leaky current, thus

$$g^{1}_{i,j}\left({V_S-V^{j}_{rest}}\right)={g^{j}_{L}} ~{V^{PSP}_{i,j}}, $$

and

$$g^{1}_{i,j}={g^{j}_{L}} ~\frac{V^{PSP}_{i,j}}{V_S-V^{j}_{rest}}. $$

(54)

Now we insert Eqs. (53) and (54) into (50), integrate and obtain

$$ {\overline{g}_{S,i,j}} =g^{j}_L ~\frac{V^{PSP}_{i,j}}{V_S-V^{j}_{rest}} \frac{\pi ~d_{i,j}^{2}~P^{hit}_{i,j} ~\rho_{i} }{1-\exp\left(-(a_{i,j}/d_{i,j})^{2}\right)}. $$

(55)

Unfortunately, the estimation is rather sensitive to the parameters $a_{i,j}$, $d_{i,j}$ which are rarely given in the papers presenting the pair registration data. Moreover, the precision of the measured voltage difference in denominator for IPSPs is low. In our calculations we set $a_{i,j}=d_{i,j}=100\mu$m, $\rho _{E4}=\rho _{E23}=2 \cdot 10^{6}\textit {cm}^{-2}$ and $\rho _{I4}=\rho _{I23}=0.5 \cdot 10^{6}\textit {cm}^{-2}$. From Thomson et al. (2002) and (Thomson and Lamy 2007) we got the following values: $P^{hit}_{E4,E4}=0.18$, $P^{hit}_{E4,I4}=0.2$, $P^{hit}_{I4,E4}=0.5$, $P^{hit}_{I4,I4}=0.5$, $P^{hit}_{E23,E23}=0.1$, $P^{hit}_{E23,I23}=0.25$, $P^{hit}_{I23,E23}=0.28$, $P^{hit}_{I23,E23}=0.5$, $P^{hit}_{E23,I4}=0.19$, $P^{hit}_{E4,E23}=0.1$, $P^{hit}_{E4,I23}=0.1$, $P^{hit}_{I4,I23}=0.27$, $P^{hit}_{E23,E4}=P^{hit}_{I23,E4}=P^{hit}_{I23,E4}=P^{hit}_{I23,I4}=0$, $V_{ampa}=0$, $V_{gaba}=-80$ mV, $V^{E4}_{rest}=V^{I4}_{rest}=V^{E23}_{rest}=V^{I23}_{rest}=-65$ mV, $V^{PSP}_{E4,E4}=1.1$ mV, $V^{PSP}_{E4,I4}=3.7$ mV, $V^{PSP}_{I4,E4}=1.0$ mV, $V^{PSP}_{I4,I4}=1.5$ mV, $V^{PSP}_{E23,E23}=1.4$ mV, $V^{PSP}_{E23,I23}=2.0$ mV, $V^{PSP}_{I23,E23}=1.0$ mV, $V^{PSP}_{I23,E23}=1.2$ mV, $V^{PSP}_{E23,I4}=1.0$ mV, $V^{PSP}_{E4,E23}=1.9$ mV, $V^{PSP}_{E4,I23}=1.2$ mV, $V^{PSP}_{I4,I23}=0.4$ mV. The obtained matrix of ratios ${\overline {g}_{S,i,j}}/g^{0}_{tot}$ for AMPA and GABA-mediated connections is shown in Fig. 17. In order to provide stability of the population model and tune it to the whole set of experimental data mentioned in the Results section, the matrix has been corrected, keeping the structure of dominance of the main connections. The final matrix is given in Table 1.

Fig. 17

Scheme of connections. The numbers are the maximum AMPA and GABA-mediated conductances scaled by the input conductances of postsynaptic cells

Appendix F: Firing rate dependence on voltage rate

As noted in Section 3.5, the rate of reaching the spiking threshold by all neurons of a population depends on the rate of mean voltage and thus because in the typical response the maximum voltage rate precedes the peak of the voltage, then the peak of the rate precedes the peak of voltage too. This effect was observed in experiments (Benucci et al. 2009). Membrane voltage and spikes in cat’s visual cortex were measured by the voltage-sensitive dyes and electrode arrays.

In order to clarify how does this issue relates to visual respones, let’s consider a simple model describing the calculation of the population firing rate $\nu (t)$ if known voltage $U(t)$ (and its derivative $dU/dt$). We apply this model to experimental voltage transient response to a short (32 ms) optimally oriented stimulus, presented in figure 2a in Benucci et al. (2009), and obtain the firing rate prediction to be compared with the experimental firing rate from that figure. Then, we estimate the spatio-temporal voltage response as a convolution of the experimental orientation tuning curve for voltage (figure 1d from Benucci et al. (2009)) with the shape of the voltage evolution profile and calculate the prediction for firing rate orientation curve to be compared with one from figure 1f from Benucci et al. (2009).

Given a mean voltage of a population, the firing rate is calculated according to Eq. (1) with the initial condition $\rho (t,t^{*}=\infty )=1$. The membrane voltage distribution in $t^{*}$-space was assumed to be an exponent saturated up to the measured voltage $V_{exp}(t)$, according to the leaky model, i.e. $U(t,t^{*})=(1-\exp (-t^{*}/\tau _{m}))~V_{exp}(t)$. The parameters were $\tau _{m}=20$ms, ${V_{th}}=10$mV, $\sigma _V=3$mV and the voltage maximum was supposed to be 15 mV. The solution is shown in Fig. 18(a). Similar to the experiment, the firing rate peaked earlier than the membrane potential. This property of the model refers to the fact that the source term in the equation for $\rho (t,t^{*})$, Eq. (1), depends on the hazard function H which is explicitly dependent on U as well as $dU/dt$. Further, after a bump of the spiking response a second weaker component is observed in the experiment and the model, which is explained by a contribution of secondary spikes occurring after some relative refractory period determined by the membrane time constant. Note, that the mentioned characteristics of the firing can hardly be described by a conventional firing rate model.

Fig. 18

Firing rate response to a visual bar stimulation, predicted from the voltage response experimentally registered in Benucci et al. (2009). a the time evolution of the maximum response for experimental voltage (figure 2a from Benucci et al. (2009)) and modeled firing rate. b the voltage (figure 1d from Benucci et al. (2009)) and firing rate profiles across preferred orientation relative to stimulus

According to Benucci et al. (2009), tuning width (half-width at half-height) for membrane potential remained fairly constant during the course of the responses and averaged 31$^{o}$. For spikes, the tuning width decreased from close to 30$^{o}$ to almost 20$^{o}$, before broadening again. Similar effect is seen in the model. Thus, the effect of transient sharpening can be explained by the simple population model of the leaky integrate-and-fire neurons with a noise.

Appendix G: Transient response to a small spot of moving gratings

An example of a simulated transient cortical response to visual stimulation by a small spot of moving gratings is given in Fig. 19. The transient response shown in Fig. 19(c) can be referred to a response of a representative neuron of a layer 2/3 excitatory population which gives a maximum response to the vertical gratings. The modeled response can be compared to the response of a simple cell recorded in area 17 of a cat, presented in figure 14B of Chavane et al. (2011). The activity at the remote site (circle) is weak, appears without a significant latency and is determined by monosynaptic excitatory population excitation and di-synaptic transition by the horizontal connections.

Fig. 19

Transient response to a small spot of moving gratings of vertical (during initial 500 ms) and horizontal (after 500 ms) orientations. a one shot of the stimulus. b the spatial distribution of the firing rate at four time moments. The extreme black and white colors correspond to 0 and 50 Hz, respectively. The cortical domain size was 2.9 mm, the spatial gratings wavelength retinonopically projected to 2 mm in the cortex. The grating frequency was 4 Hz. The stimulus was given at t = 0, after a gray screen of the same luminance as the background. The white rectangular contour bounds the region receiving the input with gratings. c the voltage and firing rate responses of excitatory population of the layer 2/3 at the locations marked by small white square and circle in the first frame in (b)