fixed panel reference

nonlinearbaseline.tex

@@ -493,7 +493,7 @@ P-units fire action potentials probabilistically phase-locked to the self-genera
 
 Noise stimuli, here random amplitude modulations (RAM) of the EOD (\subfigref{fig:punit}{C}, top trace, red line), have been commonly used to characterize stimulus-driven responses of sensory neurons using transfer functions (first-order susceptibility), spike-triggered averages, or stimulus-response coherences. Here, we additionally estimate from existing recordings the second-order susceptibility to quantify nonlinear encoding. P-unit spikes align more or less clearly with fluctuations in the RAM stimulus. A higher stimulus intensity, here a higher contrast of the RAM relative to the EOD amplitude (see methods), entrains the P-unit response more clearly (light and dark blue for low and high contrast stimuli, respectively, \subfigrefb{fig:punit}{C}). Linear encoding, quantified by the first-order susceptibility or transfer function, \eqnref{linearencoding_methods}, is similar for the two RAM contrasts in this low-CV P-unit (\subfigrefb{fig:punit}{D}), as expected for a linear system. The first-order susceptibility is low for low frequencies, peaks in the range below 100\,Hz and then falls off again \citep{Benda2005}.
 
-The second-order susceptibility, \eqnref{chi2},  quantifies for each combination of two stimulus frequencies $f_1$ and $f_2$ the amplitude and phase of the stimulus-evoked response at the sum $f_1 + f_2$ (and also the difference, \subfigrefb{fig:lifsuscept}{A}). Large values of the second-order susceptibility indicate stimulus-evoked peaks in the response spectrum at the summed frequency that cannot be explained by linear response theory. Similar to the first-order susceptibility, the second-order susceptibility can be estimated directly from the response evoked by a RAM stimulus that stimulates the neuron with a whole range of frequencies simultaneously (\subfigsref{fig:punit}{E, F}). For LIF and theta neuron models driven in the supra-threshold regime, theory predicts nonlinear interactions between the two stimulus frequencies, when the two frequencies $f_1$ and $f_2$ or their sum $f_1 + f_2$ exactly match the neuron's baseline firing rate $r$ \citep{Voronenko2017,Franzen2023}. Only then, additional stimulus-evoked peaks appear in the spectrum of the spiking response that would show up in the second-order susceptibility as a horizontal, a vertical, and an anti-diagonal line (\subfigrefb{fig:lifsuscept}{B}).
+The second-order susceptibility, \eqnref{chi2},  quantifies for each combination of two stimulus frequencies $f_1$ and $f_2$ the amplitude and phase of the stimulus-evoked response at the sum $f_1 + f_2$ (and also the difference, \subfigrefb{fig:lifsuscept}{B}). Large values of the second-order susceptibility indicate stimulus-evoked peaks in the response spectrum at the summed frequency that cannot be explained by linear response theory. Similar to the first-order susceptibility, the second-order susceptibility can be estimated directly from the response evoked by a RAM stimulus that stimulates the neuron with a whole range of frequencies simultaneously (\subfigsref{fig:punit}{E, F}). For LIF and theta neuron models driven in the supra-threshold regime, theory predicts nonlinear interactions between the two stimulus frequencies, when the two frequencies $f_1$ and $f_2$ or their sum $f_1 + f_2$ exactly match the neuron's baseline firing rate $r$ \citep{Voronenko2017,Franzen2023}. Only then, additional stimulus-evoked peaks appear in the spectrum of the spiking response that would show up in the second-order susceptibility as a horizontal, a vertical, and an anti-diagonal line (\subfigrefb{fig:lifsuscept}{B}).
 
 For the example P-unit, we observe a ridge of elevated second-order susceptibility for the low RAM contrast at $f_1 + f_2 = r$ (yellowish anti-diagonal, \subfigrefb{fig:punit}{E}). This structure is less prominent for the stronger stimulus (\subfigref{fig:punit}{F}). Further, the overall level of the second-order susceptibility is reduced with increasing stimulus strength. To quantify the structural changes in the susceptibility matrices we projected the susceptibility values onto the diagonal (white dashed line) by averaging over the anti-diagonals (\subfigrefb{fig:punit}{G}). At low RAM contrast this projection indeed has a distinct peak close to the neuron's baseline firing rate (\subfigrefb{fig:punit}{G}, dot on top line). For the higher RAM contrast this peak is much smaller and the overall level of the second-order susceptibility is reduced (\subfigrefb{fig:punit}{G}). The reason behind this reduction is that a RAM with a higher contrast is not only a stimulus with an increased amplitude, but also increases the total noise in the system. Increased noise is known to linearize signal transmission \citep{Longtin1993, Chialvo1997,  Roddey2000, Voronenko2017} and thus the second-order susceptibility is expected to decrease.