updating comments

susceptibility1.tex

@@ -518,7 +518,7 @@ The smaller \fdiff{} power spectral peak observed during pure sine-wave stimulat
 %\eqnref{response_modulation}
 
 \subsection{Low CVs and weak stimuli are associated with strong nonlinearity}
-The nonlinear effects shown for single cell examples above are supported by the analysis of the pool of 222 P-units and 45 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the nonlinearity index \nli{}, see \eqnref{eq:nli_equation}. The \nli{} characterizes the peakedness of the projections onto the diagonal of the \suscept{} matrices (e.g. \subfigref{cells_suscept}{G} at \fbase{}). It assumes high values when the \fbase{} peak in the projected diagonal is pronounced relative to the median of the diagonal and is small when there is no distinct peak. The P-unit population has higher \nli{} values up to three that depend weakly on the CV of the ISI distribution under baseline condition. Cells with lower baseline CVs have the tendency to exhibit a stronger nonlinearity than those that have high CVs (Pearson's $r=-0.16$, $p<0.01$). The two example P-units shown before (\figrefb{cells_suscept} and \figrefb{cells_suscept_high_CV}) are highlighted with dark circles in \subfigrefb{data_overview_mod}{A, C}. The stimulus strength plays an important role. Several of the recorded neurons contribute with two dots to the data as their responses to the same RAM stimulus but with different contrasts were recorded. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{data_overview_mod}{A}, see methods). Since P-units are heterogeneous in their susceptibility to the stimulus\citealp{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{data_overview_mod}{C}) a negative correlation is observed showing that cells that are strongly driven by the stimulus show less nonlinearity while those that are only weakly driven show higher nonlinearities. Whether or not a cell responds nonlinearly thus depends on both, the baseline CV (i.e. the internal noise) and the response strength. 
+The nonlinear effects shown for single cell examples above are supported by the analysis of the pool of 222 P-units and 45 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the nonlinearity index \nli{}, see \eqnref{eq:nli_equation}. The \nli{} characterizes the peakedness of the projections onto the diagonal of the \suscept{} matrices (e.g. \subfigref{cells_suscept}{G} at \fbase{}). It assumes high values when the \fbase{} peak in the projected diagonal is pronounced relative to the median of the diagonal and is small when there is no distinct peak. The P-unit population \nli{} values depend weakly on the CV of the ISI distribution under baseline condition. Cells with lower baseline CVs have the tendency to exhibit a stronger nonlinearity than those that have high CVs (Pearson's $r=-0.16$, $p<0.01$). The two example P-units shown before (\figrefb{cells_suscept} and \figrefb{cells_suscept_high_CV}) are highlighted with dark circles in \subfigrefb{data_overview_mod}{A, C}. The stimulus strength plays an important role. Several of the recorded neurons contribute with two dots to the data as their responses to the same RAM stimulus but with different contrasts were recorded. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{data_overview_mod}{A}, see methods). Since P-units are heterogeneous in their susceptibility to the stimulus\citealp{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{data_overview_mod}{C}) a negative correlation is observed showing that cells that are strongly driven by the stimulus show less nonlinearity while those that are only weakly driven show higher nonlinearities. Whether or not a cell responds nonlinearly thus depends on both, the baseline CV (i.e. the internal noise) and the response strength. 
 %(Pearson's $r=-0.35$, $p<0.001$)
 %In a P-unit population where each cell is represented not by several contrasts but by the lowest recorded contrast, \nli{} significantly correlates with the CV during baseline ($r=-0.17$, $p=0.01$), the response modulation ($r=-0.35$, $p<0.001$) and \fbase{} ($r=-0.32$, $p<0.001$).%, $\n{}=221$*, $\n{}=221$******, $\n{}=221$
 
@@ -821,29 +821,33 @@ From each simulation run, the first second was discarded and the analysis was ba
 % The second-order susceptibility in the model was calculated with \eqnref{eq:susceptibility}, resulting in matrices as in \figrefb{model_and_data} and \figrefb{model_full}. For this, the model neuron was presented the input $x(t)$ for 2\,s, with the first second being dismissed as the transient. The second-order susceptibility calculation was performed on the last second, resulting in a frequency resolution of 1\,Hz. 
 
 \subsection{Model noise split into a noise and a stimulus component}\label{intrinsicsplit_methods}
-According to the Novikov-Furutsu Theorem \citealp{Novikov1965, Furutsu1963} the total noise of a LIF model ($\xi$) can be split up into several independent noise processes with the same correlation function. Here we split the internal noise into two parts: (i) One part is treated as a driving input signal ($\xi_{signal} = c_{signal} \cdot \xi$) and used to calculate the cross-spectra in \eqnref{eq:crosshigh} and (ii) the remaining noise ($\xi_{noise} = (1-c_{signal})\cdot\xi$) that is treated as pure noise. In this way the effective signal-to-noise ratio can be increased while maintaining the total noise in the system. In our case the model has a carrier (the fish's self-generated EOD) and we thus want to drive the model with an amplitude modulation stimulus.
+According to the Novikov-Furutsu Theorem \citealp{Novikov1965, Furutsu1963} the total noise of a LIF model ($\xi$) can be split up into several independent noise processes with the same correlation function. Here we split the internal noise into two parts: (i) One part is treated as a driving input signal ($\xi_{signal} =  \sqrt{\rho \, 2D \,c_{signal}} \cdot \xi(t) $) and used to calculate the cross-spectra in \eqnref{eq:crosshigh} and (ii) the remaining noise ($\xi_{noise} = \sqrt{2D \, c_{noise}}\,\xi(t)$) that is treated as pure noise. In this way the effective signal-to-noise ratio can be increased while maintaining the total noise in the system. $\rho$ a scaling factor that compensates (see below) for the signal transformations the amplitude modulation stimulus undergoes in the model, i.e. the threshold and the dendritic lowpass. In our case the model has a carrier (the fish's self-generated EOD) and we thus want to drive the model with an amplitude modulation stimulus
 
+%(1-c_{signal})\cdot\xi$c_{noise} = 1-c_{signal}$
+%c_{signal} \cdot \xi
 \begin{equation}
     \label{eq:ram_split}
-    x(t) = (1+ \sqrt{\rho \, 2D \,c_{signal}} \cdot \xi(t)) \cdot \cos(2\pi f_{EOD} t)
+    x(t) = (1+ \xi_{signal}) \cdot \cos(2\pi f_{EOD} t)
 \end{equation}
-with $\rho$ a scaling factor that compensates (see below) for the signal transformations the amplitude modulation stimulus undergoes in the model, i.e. the threshold and the dendritic lowpass.
+
 \begin{equation}
   \label{eq:Noise_split_intrinsic_dendrite}
   \tau_{d} \frac{d V_{d}}{d t} = -V_{d}+  \lfloor x(t) \rfloor_{0}
 \end{equation}
 
 
+\begin{equation}
+  \label{eq:Noise_split_intrinsic}
+  \tau_{m} \frac{d V_{m}}{d t}  = - V_{m} + \mu + \alpha V_{d} - A + \xi_{noise} 
+\end{equation}
+% das stimmt so, das c kommt unter die Wurzel!
+
 %\begin{equation}
 %  \label{Noise_split_intrinsic}
 %  V_{m_{i+1}}  = V_{m_i} + \left(-V_{m_i} + \mu + \alpha V_{d_i} - A_i + \sqrt{\frac{2D c_{noise}}{\Delta t}}\mathcal{N}(0,\,1)_i\right) \frac{\Delta t}{\tau_m}
 %\end{equation}
 
-\begin{equation}
-  \label{eq:Noise_split_intrinsic}
-  \tau_{m} \frac{d V_{m}}{d t}  = - V_{m} + \mu + \alpha V_{d} - A + \sqrt{2D \, c_{noise}}\,\xi(t)
-\end{equation}
-% das stimmt so, das c kommt unter die Wurzel!
+
 
 A big portion of the total noise was assigned to the signal component ($c_{signal} = 0.9$) and the remaining part to the noise component ($c_{noise} = 0.1$, \subfigrefb{flowchart}\,\panel[iii]{C}). For the application of the Novikov-Furutsu Theorem \citealp{Novikov1965, Furutsu1963} it is critical that both components add up to the initial 100\,$\%$ of the total noise. This is easily achieved in a model without a carrier if the condition $c_{signal}+c_{noise}=1$ is satisfied. The situation here is more complicated. To compensate for the transformations the signal undergoes before it enters the LIF core, $\xi_{signal}(t)$ was scaled up by the factor $\rho$ (\eqnref{eq:ram_split}, red in \subfigrefb{flowchart}\,\panel[i]{D}). $\rho$ was found by bisecting the space of possible $\rho$ scaling factors by minimizing the error between the baseline CV (only carrier) and the CV during stimulation with noise split.