In the present study, we have achieved a theoretical and numerical sensitivity analysis aimed at systematically assessing the impact of voltage-gated maximal conductance modifications on the
. We focus on the
to explore the still largely obscure effects of IP on neuronal rate coding, which represents the functionally relevant regime of many neurons that share type I excitability, and is extensively employed experimentally to assess IP.
Our study leads to the general principle that the effect of maximal conductance plasticity on firing rate is governed by two additive terms which separately affect the threshold and the inverse gain of the
. For a given maximal conductance change, these effects are weighted by two parameters, the threshold and the inverse gain sensitivities to the maximal conductance. These sensitivities are themselves independent of the maximal conductance. Rather, they reflect how kinetics (i.e. qualitative properties) of the conductance modulate the way maximal conductance changes affect the
. As anticipated by our initial theoretical analysis, an extensive exploration of sensitivities in the parameter space of HH models systematically demonstrated two contiguous, marginally overlapping domains of elevated threshold or inverse gain sensitivities in the
On the one hand, conductance activating “sub-threshold”, independent of the occurrence of spikes, display high threshold sensitivities, i.e. changes in their maximal conductance strongly affect the
threshold of the neuron. Consistent, we have shown in analytically IAF neuron models that the threshold sensitivity solely depends on the activation level
, the effective AP threshold, which does not depend on firing frequency. Thus, IP in these cases affects the
independently of firing frequency, i.e. it shifts the
so that the threshold but not the gain is modified. Besides, it arises from the dependence of threshold sensibility on
that conductance with very different activation functions but sharing the same
values present the same threshold sensitivity. By contrast, conductance with steep activation functions shifted by only a few mV can display extremely different threshold sensitivities. Thus, our results offer the possibility of estimating threshold sensitivity of real conductance assuming that their infinite activation and
are known with reasonable accuracy.
On the other hand, conductance activating “supra-threshold”, i.e. in correlation with spikes, present high inverse gain sensitivities, i.e. changes in their maximal conductance strongly affect the
gain. We have shown that inverse gain sensitivity depends on two mechanisms related to the activation relaxation following APs, and the activation buildup preceding APs. Interestingly, these two mechanisms rely respectively on the difference between activation levels 1) attained at the end of the spike and at the reset potential and 2) the reset potential and the threshold potential. Hence, conductance displaying large inverse gain sensitivities have half-activation potentials situated above the reset potential and below ~−20 mV and small e-fold slopes. Moreover, the pre-and post-spike IAF theories we have studied indicate that conductance with fast activation kinetics privilege the buildup effect while slower kinetics of activation favor the deactivation mechanism. As for the threshold sensitivity, we furthermore found that a precise knowledge of biophysical parameters can be crucial in estimating the inverse gain sensitivity of conductance.
Together, these results provide a unifying framework to account for and interpret IP experiments. Indeed, we have determined that voltage-gated conductance with large threshold versus large inverse gain sensitivities can be discriminated on the basis of a simple and generic criterion, i.e. an activation of
. This criterion roughly corresponds to the classical albeit fuzzy distinction between “sub-threshold” and “supra-threshold” types of conductance. Hence, according to our results, pharmacologically identified conductance types such as the leak (IL
), the persistent (INaP
) and slowly-inactivating (INaS
) sodium, the low-threshold calcium (ICaT
), or the muscarinic (IM
) and slowly-inactivating (IKs
) potassium conductance display biophysical parameters typically situated in the domain of high threshold sensitivity. Consistently, empirical studies indicate that modifications of the persistent sodium 
, slowly inactivating potassium 
and leak conductance 
strongly correlate with large
In contrast, modifications of pharmacologically identified conductance types such as the high-threshold calcium conductance (e.g. ICaL
), calcium-activated (IAHP
) or fast-potassium potassium (IA
) conductance that are directly or indirectly activated by APs should essentially affect the
gain. Empirical evidence indicate that gain changes are indeed induced in vestibular nucleus neurons by IP of calcium-activated potassium conductance and spike triggered high-threshold R-type calcium conductance that induce their activation 
. Moreover, additional previous work has shown that the maximal conductance of calcium-activated current indeed determines the
gain in proportion to their activation time constant 
, consistent with our analysis of the post-spike IAF theory.
The dichotomy we have unraveled and which appears to beneficiate from experimental support appeals several remarks. First, our results indicate that mixed modifications of the 
do not necessarily implicate the co-regulation of two or more conductance but could simply arise from the IP of conductance situated at the overlapping of threshold and inverse gain domains. Second, parallel to the sub-/supra-threshold dichotomy, our study clearly indicates the opposition between sodium and potassium conductance of the AP, which respectively affect the threshold versus the inverse gain of the
, consistent with experimental data 
. Third, experimental data indicate that the IP of several A-type or persistent potassium conductance affects the
, while these conductance are paradoxically traditionally classified as supra-threshold because they present quite depolarized half-activation potentials (
). However, these conductance present large e-fold activation slopes (
) so they should lie in the domain of large threshold modifications. Thus, based on actual biophysical conductance parameters, the present theory correctly categorizes the IP effects of pharmacologically identified conductance, even when their apparent classification, based on the fuzzy sub/supra-threshold distinction, is misleading. Our study therefore points toward the importance of precise biophysical conductance parameters over the simple knowledge of the pharmacological conductance type in determining the rate effects of the IP of actual conductance. Finally, the validity of our theoretical results could practically be further confirmed or infirmed in detail, employing the dynamic clamp technique to experimentally measure threshold and inverse sensitivities by sampling points of interest in the biophysical parameter space. In particular, this technique could help disentangle an apparent discrepancy that we have unraveled concerning the IH
conductance. Indeed, our results predict no effect on the gain and a negative
is depolarizing (not shown), whereas several IP studies show that IH
exclusively increases the
. This may originate from indirect effects such as a decreased input resistance or putative complex interactions with other sub-threshold currents 
and geometrical factors in dendrites 
. The dynamic-clamp technique may thus separate direct and indirect effects in that case.
In addition to interpreting existing results, the present theoretical framework represents a valuable tool for experimentalists to target putative conductance involved in IP, based on the observation of
changes. Moreover, our analysis has unraveled supplementary intermediate electrophysiological observables such the effective AP threshold (Text S1
) or the ISI voltage trajectory (Text S7
), which modifications can be analyzed to refine the targeting of putative conductance of interest.
We have ascertained that the present results are robust. Indeed, shifting half-activation potentials of AP sodium and potassium currents by a few mV shifts sensitivity maps by the same amount along the
dimension but does not change their global structure (not shown). Moreover, using another model of AP conductance did not significantly change our results (not shown; 
). Furthermore, the threshold versus inverse gain sensitivity dichotomy we have demonstrated proved robust when considering net mean frequency effects that can be obtained from maximal modifications of maximal conductance preserving excitability parameters within physiological bounds (Text S16
Besides, the dichotomy we have unraveled appears to extend to the general case of voltage-dependent activation time constants, commonly encountered in real conductance. Hence, sensitivity maps obtained with voltage–dependence activation time constants (in the range 1–5 ms; not shown) were consistent with our previous understanding of sensitivities' dependence on time constants. Indeed, we found (not shown) that 1) threshold sensitivity is globally unaffected by the voltage-dependence of the activation; 2) large time constants at ISI potentials (below voltage AP threshold) increase the impact of the post-spike relaxation and delay the build-up effect, thus augmenting the post-spike mechanism and diminishing the pre-spike mechanism; 3) large time constants at spike potentials (above AP voltage threshold) diminish the activation increase during the spike, reducing the post-spike effect, but have no impact on the pre-spike mechanism.
Finally, an important question is whether the IP effects we unravel are robust in the general case where several voltage-gated conductance are present, even though exploring this issue in depth is largely out of the present scope. Actually, we have achieved a preliminary exploration suggesting that threshold and inverse gain modifications behave as the linear sum of individual conductance effects. If confirmed, this result would be noteworthy, given the degree of non-linearity commonly encountered in neurons at the level of the membrane potential or gating variables. Moreover, such linearity would open the possibility to capture complex interactions between conductance in a simple way in terms of frequency coding in neuron and neural network models.
Although robust, our results should be extended with respect to several dimensions, including 1) IP effects on spike-timing properties (e.g. higher order moments of the discharge, resonance, latency to first spike or frequency adaptation), in particular by also considering type II excitability neuron models, 2) multi-compartmental neuron models to address IP effects on dendritic integration 
, summation 
, branch computation 
and spike back-propagation 
and determine whether the sub-/supra-threshold distinction remains relevant with dendritic spikes.
This analysis complements recent analyses of parameter robustness of excitability in Hodgkin-Huxley (HH) models, using sensitivity analysis or stochastic search methods 
. Indeed, these studies assess the spontaneous dynamical regime of neurons 
or incomplete descriptions of the excitability 
, whereas our study fully quantifies the
. Moreover, they seek compensatory trade-off between conductance with specific kinetics, in the space of maximal conductance dimensions. Rather, our study is independent of the rules that actually govern IP (e.g. H/IP versus AH/IP) and explores the kinetics parameter space of a single generic model of voltage-gated conductance. Therefore, it allows evaluating independently the sensitivity of virtually any voltage-gated conductance with arbitrary kinetics and offers some insights on calcium- or second-messenger gated conductance scaling with firing frequency 
Here, we have focused on IP effects to escape the entanglement of IP effects and IP rules in empirical and theoretical studies and provide a manageable framework for the comprehensive study of IP loops. Hence, our goal is attainable by coupling the present IP effect equations with IP rules equations describing the causal mechanisms relating on-going spiking activity to conductance changes. In our mind, realistic signaling pathways models are desirable as they share the same - molecular - level of description. IP processes display gradation 
, possibly fast induction 
, long-term maintenance 
and ubiquitously involve kinase/phosphatase cycles 
so that the aKP model 
represents a natural counterpart to the present model. In the present study, we have coupled the aKP model to HH or rate coding equations to address the example of the homeostatic regulation of spontaneous discharge by the IP loop. Our results illustrate how the IP theory we have unraveled can account for the outcome of IP rules, based on the precise knowledge of conductance biophysical parameters, and provide lower computational cost and better tractability to systematically decipher the complexity of the IP loop. To model the loop, choosing autocatalytic plasticity models inducing binary switches of the plastic variable would have been clearly irrelevant 
, because homeostatic IP changes are graded 
. Similar results would be obtained using an alternative phenomenological model that produce graded changes 
. However, because it lacks activity-dependent time constant, such a model would fail - contrarily to the aKP model - to account for the slower dynamics at low electrical activity 
and faster changes under conditions of hyper-activity 
that characterize homeostatic IP experimentally. In the future, one may in a similar way realistically investigate essential issues related to IP loops at the single neuron level such as the stability problem, the emergence of dynamics of interest, information processing properties or interactions with synaptic plasticity.
Introducing these coupled equations in neural networks offers the possibility to assess the impact of IP on dynamical and computational network properties. The present results allow studying IP of real conductance with known biophysical parameters in firing rate neural networks with explicit threshold and/or gain, and spiking neural networks embedded with conductance parameters, using event-based schemes 
by taking advantage of the analytical voltage trajectories we have devised. Studying such networks would allow assessing the causal role of conductance modifications that have been correlated to various behavioral learning (e.g. trace, classical and operant conditioning, or rule learning; 
. They would also bring about gaining a global picture of the computational properties conferred by IP. Indeed, modifying the
threshold provides an additive modulation determining input selectivity, while
gain modifications operate a multiplicative modulation that scales neuronal output. These distinct forms of activity-dependent regulations should therefore participate setting very different computational properties at the level of neural networks 
with regard to dynamical regime control, information storage or history-dependent computations for instance.
As a concluding remark, the present results are independent of the regulatory processes modifying conductance parameters and thus relevant to a larger class of processes than IP, possibly including neural development 
, maturation 
, neuromodulation 
, aging 
and various neural diseases 
, in which conductance modifications represent critical cellular processes.