Detecting single-cell stimulation in a large network of integrate-and-fire neurons

Several experiments have shown that the stimulation of a single neuron in the cortex can influence the local network activity and even the behavior of an animal. From the theoretical point of view, it is not clear how stimulating a single cell in a cortical network can evoke a statistically significant change in the activity of a large population. Our previous study considered a random network of integrate-and-fire neurons and proposed a way of detecting the stimulation of a single neuron in the activity of a local network: a threshold detector biased toward a specific subset of neurons. Here, we revisit this model and extend it by introducing a second network acting as a readout. In the simplest scenario, the readout consists of a collection of integrate-and-fire neurons with no recurrent connections. In this case, the ability to detect the stimulus does not improve. However, a readout network with both feed-forward and local recurrent inhibition permits detection with a very small bias, if compared to the readout scheme introduced previously. The crucial role of inhibition is to reduce global input cross correlations, the main factor limiting detectability. Finally, we show that this result is robust if recurrent excitatory connections are included or if a different kind of readout bias (in the synaptic amplitudes instead of connection probability) is used.

Figure 1

Illustration of the two detection schemes compared in this paper. One neuron embedded in a large recurrent network representing the barrel cortex (BCN) is stimulated. We compare the readout scheme A [panel (a)], introduced in Ref. [19], with the readout scheme B [panel (b)]. In the readout scheme A, the activity of a subpopulation ${\mathcal{S}}^A$ is fed to the detector. Varying the detector sensitivity (${\theta }_{\pm }$) yields the receiver operating characteristic (ROC) curve, which describes the performance of the detector as a function of its sensitivity. The distance of the ROC curve from the diagonal for a false positive rate of 25% defines the effect size $\overline{\mathcal{Y}}$. In the readout scheme B, the readout network (RN) receives feed-forward input from the barrel cortex network. The activity of the excitatory neurons within the RN, ${\mathcal{S}}^B$, is then fed to the detector. In both cases, the parameter $\lambda$ quantifies the bias of the readout toward the set of neurons ${\mathcal{B}}_1$, the neurons one synapse away from the stimulated neuron.

Figure 2

(a) Illustration of the network representing the barrel cortex (BCN). ${\mathcal{B}}_0$ is the stimulated cell; ${\mathcal{B}}_1$ is the set of cells receiving direct input from ${\mathcal{B}}_0$; ${\mathcal{B}}_2$ is the set of cells receiving no direct input from ${\mathcal{B}}_0$. Note that the network is completely random and unstructured, and cells are grouped together only for visualization purposes. (b) Firing rate deviation of ${\mathcal{B}}_1$ and ${\mathcal{B}}_2$ in response to the stimulus, filtered with two different time steps (black curve for 1 ms, blue for 20 ms) to reduce fluctuations due to the finite number of trials (1000). The thicker red line is the theory resulting from Eqs. (9) and (11).

Figure 3

Illustration of the detector. Two trials of the readout activity $R_{\lambda }^A$ are plotted for two values of the bias parameter $\lambda$. Crossings of the threshold ${\theta }_{+}$ or of the threshold ${\theta }_{-}$ define false positive events if they occur before the stimulus onset, i.e., for $-T_w<t<0$, or correct detection events for $0<t<T_w$.

Figure 4

Detectability of single-cell stimulation does not improve if the readout network (RN) has no recurrent connections but only integrates feed-forward input from the barrel cortex network (BCN), in which a randomly selected cell (${\mathcal{B}}_0$) is stimulated. The four panels on the right side show the effect size as a function of the bias $\lambda$ for setup A (black symbols and lines) and setup B (orange symbols and lines). Detection via the ${\theta }_{+}$ detector is represented by upward pointing triangles (simulations) and continuous lines (theoretical approximation discussed in Appendix pp1). Detection via the ${\theta }_{-}$ detector is depicted with downward pointing triangles (simulation) and dashed lines (theory). In all panels, the vertical dotted line marks ${\lambda }_0$, which corresponds to no bias in the input to the detector. Closed symbols indicate statistically significant points (Fisher's exact test, $p$ value $<0.05$). Panels (a) and (c) show results for excitatory ${\mathcal{B}}_0$, while (b) and (d) show results for inhibitory ${\mathcal{B}}_0$. In panels (a) and (b) the mean input is set such that the spontaneous firing rate of ${\mathcal{S}}^B$ is equal to that of the BCN ($\approx 2\mathrm{Hz}$). In panels (c) and (d) the firing rate of the readout is higher ($\approx 15\mathrm{Hz}$) to make the readout activity more Gaussian.

Figure 5

Large enhancement in the detectability of single-cell stimulation owing to local inhibition. Here, the readout network (RN) consists of the population ${\mathcal{S}}^B$ and the local inhibitory population $\mathcal{I}$. Both ${\mathcal{S}}^B$ and $\mathcal{I}$ receive feed-forward input from the barrel cortex network (BCN), in which a random cell is stimulated, and inhibitory input from $\mathcal{I}$. Panels on the right side show the effect size as a function of the bias for setup A, setup B, and both detector types ${\theta }_{\pm }$. Color code and symbols are as in Fig. 4. Panels (a) and (c) show results for excitatory ${\mathcal{B}}_0$, while panels (b) and (d) show results for inhibitory ${\mathcal{B}}_0$. In panels (a) and (b) ${\lambda }_e$ (the connection bias from the BCN to ${\mathcal{S}}^B$) is varied while ${\lambda }_A={\lambda }_e$ and ${\lambda }_i={\lambda }_0$. In panels (c) and (d) ${\lambda }_i$ (the connection bias from the BCN to $\mathcal{I}$) is varied while ${\lambda }_A={\lambda }_i$ and ${\lambda }_e={\lambda }_0$.

Figure 6

Fully recurrent E-I network also enhances the detectability of the single-cell stimulation. Here, the readout network (RN) consists of an excitatory population ${\mathcal{S}}^B$ and an inhibitory population $\mathcal{I}$, both recurrently connected. The RN receives recurrent excitatory input from ${\mathcal{S}}^B$ (75% of the total recurrent input) and feed-forward excitatory input from the barrel cortex network (BCN) (25% of total input). Panels (a) and (c) show results for excitatory ${\mathcal{B}}_0$, while panels (b) and (d) show results for inhibitory ${\mathcal{B}}_0$. In panels (a) and (b) the effect size as a function of the connection bias for excitatory feed-forward projections ${\lambda }_e$ (as in the previous cases) is plotted (${\lambda }_i={\lambda }_0$ and ${\lambda }_A={\lambda }_e$), while in panels (c) and (d) the bias $\beta$ represents the relative strength of connections from ${\mathcal{B}}_1$ (see text). Color code and symbols are as in Figs. 4 and 5.

Figure 7

Qualitative effects do not depend on the choice of the detector. Effect size obtained from the detector used in this study and in Ref. [19], applied to the case considered in Sec. 2b. The detector with two symmetric thresholds (orange circles) detects both above and below ${\lambda }_0$ but the effect size is smaller in magnitude. In panel (a) are plotted results for a fixed false positive rate, as done in the main text; in panel (b) we plot results for an optimized threshold, as done in Ref. [19]. With this definition the effect size cannot be negative. Otherwise, for positive effect size it can be seen that optimizing the threshold has a very small effect as comparison to panel (a) reveals. In (b) we omit indicating the statistical significance (see Appendix pp3).

Figure 8

Simplified detection experiment without signal in order to mimic catch trials. Distribution of effect size (here only due to finite-size fluctuations) and $p$ value calculated with Fisher's test for three ways of choosing the threshold: fixed threshold [(a), (b)], threshold corresponding to fixed false positive rate [(c), (d)], and optimal threshold [(e), (f)].