https://orcid.org/0000-0002-7098-4768
Figures
Abstract
In this article we demonstrate a general protocol for constructing systematically experimental steady-state bifurcation diagrams for electrophysiologically active cells. We perform our experiments on entorhinal cortex neurons, both excitatory (pyramidal neurons) and inhibitiory (interneurons). A slowly ramped voltage-clamp electrophysiology protocol serves as closed-loop feedback controlled experiment for the subsequent current-clamp open-loop protocol on the same cell. In this way, the voltage-clamped experiment determines dynamically stable and unstable (hidden) steady states of the current-clamp experiment. The transitions between observable steady states and observable spiking states in the current-clamp experiment provide partial evidence for stability and bifurcations of the steady states. This technique for completing steady-state bifurcation diagrams in a model-independent way expands support for model validation to otherwise inaccessible regions of the phase space. Overlaying the voltage-clamp and current-clamp protocols leads to an experimental validation of the classical slow-fast dissection method introduced by J. Rinzel in the 1980s and routinely applied ever since in order to analyse slow-fast neuronal models. Our approach opens doors to observing further complex hidden states with more advanced control strategies, allowing to control real cells beyond pharmacological manipulations.
Author summary
We demonstrate a general protocol for constructing systematically experimental steady-state bifurcation diagrams of neurons. We test it with both excitatory (pyramidal neurons) and inhibitory (interneurons) cells from the entorhinal cortex of rats. A slowly ramped voltage-clamp electrophysiology protocol serves as closed-loop feedback controlled experiment for the subsequent current-clamp open-loop protocol on the same cell. In this way, the voltage-clamped experiment determines dynamically stable and unstable (hidden) steady states of the current-clamp experiment. The transitions between observable steady states and observable spiking states in the current-clamp experiment provide partial evidence for stability and bifurcations of the steady states. This technique for completing steady-state bifurcation diagrams in a model-independent way expands support for model validation to otherwise inaccessible regions of the phase space. We explain why the method works by using multiple-timescale dynamics arguments.
Citation: Amakhin D, Chizhov A, Girier G, Desroches M, Sieber J, Rodrigues S (2025) Observing hidden neuronal states in experiments. PLoS Comput Biol 21(12): e1013748. https://doi.org/10.1371/journal.pcbi.1013748
Editor: Matthias Helge Hennig, University of Edinburgh, UNITED KINGDOM OF GREAT BRITAIN AND NORTHERN IRELAND
Received: May 12, 2025; Accepted: November 15, 2025; Published: December 8, 2025
Copyright: © 2025 Amakhin et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All presented data, as well as, codes to process them, are available via the following Github repository: https://github.com/serafimrodrigues/Amakhin-etal-2025/.
Funding: SR acknowledges the grant PID2023-146683OB-100 funded by MICIU/AEI375/10.13039/501100011033 and by ERDF, EU. Additionally, SR acknowledges support from Ikerbasque Foundation and the Basque Government through the BERC 2022-2025 program and by the Ministry of Science and Innovation: BCAM Severo Ochoa accreditation CEX2021-001142-S/MICIU/AEI/10.13039/501100011033. Moreover, SR acknowledges the financial support received from BCAM-IKUR, funded by the Basque Government by the IKUR Strategy and by the European Union NextGenerationEU/PRTR, as well as, support of SILICON BURMUIN no. KK-2023/00090 funded by the Basque Government through ELKARTEK Programme. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exists.
Introduction
When characterising the dynamics of nonlinear systems, one fundamental criterion for a model is if its stable states such as stationary solutions or periodic orbits match experimental observations. The ability to fit and validate models is, thus, greatly expanded by experimental tools with the capacity to unveil non-observable (sensitive or dynamically unstable) states that are otherwise inaccessible to standard measurements. The combination of observable and non-observable states gives access to an experimental equivalent of parameter-dependent families of stable and unstable states in a model, which are usually referred to as a bifurcation diagram.
This article applies a new experimental technique of using feedback control to find unstable states to electrophysiology experiments on neuronal cells. Our aim is to support systematic validation of neuron models by comparing bifurcation diagrams and observing their between-cell variability. We focus on unstable parts of input-dependent families of steady-state solutions obtained by feedback-controlled experiments and compare them with indirect evidence from standard measurements from open-loop experiments. This extends recent work of Ori et al. [1,2] constructing phase diagrams from neuronal data, and complements other approaches such as using data to verify the bifurcation structure of neuronal models [3,4], model-based data analysis [5] or parameter estimation from data [6].
Our technique is a simplification of the so-called control-based continuation (CBC) method, an approach which has been recently demonstrated in mechanical experiments [7], vibrations and buckling experiments [8], pedestrian flow experiments [9], atomic-force microscopy [10], cylindrical pipe flow simulations [11], and feasibility studies for synthetic gene networks [12,13]. Indeed, CBC is a procedure that combines feedback control [14] and pseudo-arclength continuation [15] in a model-free environment, that is, only reliant upon noisy experimental data in a closed-loop control setup. The objective is to compute experimental bifurcation diagrams, that is, both families of stable and unstable states (either stationary or periodic) together with bifurcation points joining such families. Importantly, the control is noninvasive in the sense that it vanishes once an equilibrium or periodic of the uncontrolled system has been reached; the control is decreased iteratively using a Newton’s method. In our case, the approach is simpler and does not require the use of Newton iterations since we consider a slow ramp on the control target. Hence we exploit de presence of two timescales in the experimental procedure in order to obtain directly an approximation of the steady-state bifurcation curve, and we explain why such a bifurcation curve is obtained at low cost; see below. Our approach is hence useful for both modelers and experimentalists, as it relies upon standard protocols routinely used in patch-clamp electrophysiology. It can further decipher the excitability class of a given real neuron as well as help fitting a model to data by using both current-clamp and voltage-clamp protocols.
In our electrophysiology experiments on entorhinal cortex neurons we treat the cell as an electric circuit, and apply a voltage clamp (VC) [16,17], followed by current clamp (CC). In the VC setup the electrode acts as a voltage source at the neural membrane, fixing the potential across the neural membrane, measuring the current, while the CC setup adds a fixed external current, measuring the resulting membrane potential; see the illustration for the experimental setup in Fig 1A and the S1 Text for further details. In-vivo neurons are subject to current signals that drive spiking (oscillatory) or rest (steady) states of the neural membrane potential [18]. This makes the CC setup the open-loop part of the protocol. In contrast, the VC part of our experiment is the closed-loop feedback-controlled part of the protocol because the voltage source regulates the external current to maintain the hold voltage. VC has been applied successfully to study neuronal nonlinear current-voltage relationships, so-called N-shaped I-V characteristics, which cause enhanced neuronal excitability and influence the regenerative activation of certain ionic currents (e.g., sodium) [19–23] across the neural membrane, into and out of the cell. We show that the VC protocol with a slowly varied reference voltage signal gives access to stable and unstable neuronal steady states of the neuron, which was hinted at in [24,25]. In contrast, the open loop CC protocol with a slowly varying applied current always follows stable (observable) states, driving the neuron to dynamically transition between its observable rest states and its observable spiking states.
(a), patch pipette (b), reference electrode (Ag-AgCl pellet) connected to ground (c), amplifier (e, Multiclamp 700B) with CV-7B headstage (d), AD-converter (f, National Instruments NI USB-6343) and standard PC computer (g, https://pixabay.com/vectors/computer-desktop-workstation-office-158675/). B: VC and CC protocol runs for cell 5: -curve for VC run (thin bright blue: unfiltered data with sampling time step
s, blue/red/brown: median of
over moving windows of size
steps equalling 0.2s) and
-curve for CC run (orange, mean of
over moving windows. C:
-curves for VC protocol of all 5 cells on waterfall
-axis (color coding indicates conjectured stability as indicated for (b)).
We interpret these combined experimental protocols (VC and CC) using multiple-timescale dynamics, in particular, the dissection method [26], which reveals the dynamic bifurcations in the experiment, as demonstrated in the experimental bifurcation diagram in Fig 1B. In this slow-fast framework, the states traced by the VC protocol with slow variation correspond to the steady-state experimental bifurcation diagram of the so-called fast subsystem of the mathematical model describing the protocol [26], that is, the model system with constant input current. Hence, the N-shaped I-V relation of a neuron should be seen as a S-shaped V-I bifurcation diagram, such as in Fig 1B.
Following this strategy, we demonstrate the feasibility of tracking a family of neuronal steady states (stable and unstable) via variations of reference signal and reparameterizing the obtained curve using the feedback current.
Results
The experimental neuronal bifurcation diagram in Fig 1B shows the time profiles of the CC protocol run (orange, thin) and
of the VC protocol run (bright blue, thin) for cell 5 overlaid in the (I,V)-plane. For both runs
and
were slowly increased, respectively (see Materials and Methods). After smoothing, the VC time profile is the S-shaped curve
(blue/brown/red, thick). It equals the (I,V)-characteristic of the stationary neuronal states of the CC protocol, including dynamically unstable states (brown and parts of red). The transition to stable spiking states is compatible with (Hodgkin) class-I excitability, however we do not have sufficient data to distinguish the different classes of excitability. See Fig B in S1 Text for how the interspike intervals depend on
.
The dynamical stability and instability of stationary states is inferred based on two pieces of evidence: (i) the negative slope of the -curve (brown) after smoothing over a moving window with larger size (
steps equaling = 0.75s in Fig 1C), or (ii) by the presence of slow-fast oscillations (neuronal firing) in the CC run at
equalling the
(red). Criterion (i) implies instability for topological reasons. Criterion (ii) provides only circumstantial evidence. In Fig 1B (cell 5) the darker shading of the (orange-colored) CC run indicates slow dynamics as identified by the norm
being less than a fixed threshold (after smoothing of
and
, see S1 Text for detailed definition). This shading shows that the firing oscillations in Fig 1B spend most time near their voltage minimum
for the respective
. Fig 5C and 5E show an embedding of the firing oscillations of cell 5 into the
-plane, and a zoom near
pA. The projection and zoom indicate that the firing oscillation passes slowly near the presumed fixed point, but leaves its neighborhood again. At the low-current end of the
-curve, the CC protocol for cell 5 shows no firing oscillations (recall that
is ramped up) in Fig 1B. Thus, the part of the
curve between fold and presence of slow-fast oscillations is labelled “undetermined” (colored yellow) as the combination of single-ramp VC and CC protocol do not provide evidence for or against stability of this part. We do not label the “transition” from stability label “undetermined” to “unstable” as a Hopf bifurcation as the slow parts of the firing oscillations are approximately 20 mV below the equilibrium indicated by the VC run. So, it is unclear if (and where precisely) a change of stability occurs between
pA and the fold of steady states at
pA.
The stability boundaries of the stationary states are labelled as bifurcations in Fig 1B. The change of stability near the disappearance of stable spiking states at is labelled as a Hopf bifurcation. We observe that for
small-amplitude oscillations are visible, which emerge from relaxation type oscillations with a slow phase near the stationary state for
. Fig I in S1 Text shows a zoom of Fig 1B near
. These features are typical for a singular Hopf bifurcation [27] as encountered also in model simulations for excitatory neurons. The fold points of the
-curve are saddle-node (fold) bifurcations. Fig 1C shows the stationary-state curves with their inferred dynamical stability for all 5 cells (see S1 Text for CC run time profiles used to partially infer stability of cells 1–4). The cells are vertically ordered and numbered according to depth of the S-shape, determining if they fall into the category of class-I or class-II neurons. We use the same convention as in Fig 1B for parts of the curve where we have partial evidence for instability: no bifurcation is indicated at transitions between stability labels “undetermined” and “unstable”. Fig 1C demonstrates wide variability in steady-state curve shape among cells of nominally same function.
The method described in this work is versatile and uses only standard electrophysiological procotols. Hence, it can be applied to any neuron type. The experimental bifurcation diagrams in Fig 2A illustrate this point. Panel A shows the results of the VC and CC protocols applied to an inhibitory neuron from the entorhinal cortex while panel B shows the results of same protocols applied to a class-II neuron. The steady-state curve in Fig 2A is more complicated than for PY neurons’ recordings shown in other figures. However it is compatible with a class-I excitable neuron. Since the steady-state curve in Fig 2A has multiple folds, there are some parts of the branch where we cannot determine stability safely from topology (that is, by checking that ), or from the near-by embedded trajectories of relaxation oscillations. Similar to Fig 1 we have colored the parts of the steady-state curve in Fig 2A with undetermined stability yellow, while keeping the part for which instability is suggested by existence of relaxation oscillations in red. For the range of
between 43 pA and 60 pA during the CC protocol in Fig 2A we observe a stable steady state
response, with intermittent voltage spikes typical for an excitable stable steady state near a singular Hopf bifurcation subject to noise [28]. For this reason we label the low-
onset of large-amplitude relaxation oscillations in Fig 2A also as a Hopf bifurcation. A potential alternative mechanism for the sudden transition of firing oscillations to steady state near
(
pA for Fig 2A,
pA for Fig 2B) is that the firing oscillations experience a saddle-node of periodic orbits. This would imply a bistability between firing and steady state for
, a scenario that has been observed near depolarization blocks in models of dopaminergic neurons by [29]. Our illustrative examples of mathematical neuron models shown in Figs 3A and 4 have such a small region of bistability (Fig 3A near
and Fig 4 near
pA) for the parameters used in the computations (see Figs F, G and H in S1 Text for numerical bifurcation diagrams).
B: Experimental bifurcation diagram for a class-II PY neuron from the same region. Protocols identical and color coding to Fig 1B.
Protocol as described for Fig 1B applied to A: a class-I Morris-Lecar neuron model (10) [33] as example of excitatory cell, and B: a class-I Wang-Buzsáki neuron model (12) [34] as example of inhibitory interneuron; see (10) and (12) for differential equations and Tables 1 and 2 for parameter values. The two-dimensional fast subsystem has a S-shaped steady-state curve satisfying steady-state conditions (4). The steady-state I–V curve (4) and the (multi-color) S-shaped curve from the VC protocol (2) are indistinguishable throughout the range of input currents . The orange curve resulting from the CC protocol is very close to S0 and the VC protocol near its dynamically stable parts. See Figs F and G in S1 Text for numerical bifurcation diagrams.
See (10) for differential equations and Table 1 for parameter values and Fig H in S1 Text for numerical bifurcation diagram.
We observe that the steady-state curves for the CC run and the VC run slightly deviate from each other, which we attribute to a drift in cell properties. There are examples of scenarios with multiple folds in the steady-state curve in the literature, in particular with low-threshold spiking neurons [30].
Analysis
To see why VC-run time profiles approximate the experimental bifurcation diagram with unstable stationary states of neurons in CC protocol, we use the formalism of multiple-timescale dynamical systems. Superimposing the data from VC and CC protocol also gives the first experimental illustration of slow-fast dissection. The effect of the respective clamps can be understood in a general class of conductance-based models for the neuron,
describing the current balance across the neuron’s membrane. The membrane potential is V, are the currents across different voltage-gated ion channel types and
is the external current. Each channel type j has a set of associated gating variables xj, which are possibly vectors of length nj if the channel gate has both activation and inactivation states, with steady-state gating functions
(also of size nj) and relaxation times
(a diagonal
matrix). The observed dynamic effects such as oscillations (firing/spiking) and negative-slope (I,V) characteristics are determined by these channel coefficients Ij,
and
that are traditionally obtained by parameter fitting from VC experiments, a difficult and ill-posed problem, as gating variables xj are not directly measured [31].
The VC and CC protocols use different mechanisms for generating . The VC protocol is a closed loop where a voltage source regulates
with high gain
to achieve the slowly varying hold voltage
at the voltage source for general model (1), measuring
:
which turns general model (1) with VC protocol (2) into a multiple-timescale dynamical system with fast state variables
(where
is the overall number of gating variables) and one slow state variable
, corresponding to the feedback reference signal [32]. The speed at which
varies is
with
mV/ms, where we extract the dimensionless small factor
.
In contrast, the CC protocol holds , measuring the generated voltage
, thus, corresponding to an open-loo system, permitting e.g. the spiking seen in Fig 1B:
The applied hold current is varied slowly at speed
with
and
pA/ms. General model (1) with CC protocol (3) is also a slow-fast system with
fast variables
and 1 slow variable
. The gain
in VC protocol (2) is limited by the imperfect conductance across the non-zero spatial extent of the membrane. Even though the conductance-based model (1) is for the potential V across the entire membrane and only
at the clamp is measured, we approximate
for the membrane potential, and
for the external current in (1).
Following a classical multiple-timescale approach, we consider the limit of general model (1) with VC protocol (2), which corresponds to its
-dimensional fast subsystem (1), with
, where
is now treated as a parameter. For fixed
and voltages in the range
mV of interest, model (1) with VC protocol (2) has only stable steady states (no limit cycles, that is, no neuronal spikes); see S1 Text for details. For a fixed hold voltage
, the steady states of system (1), (2) with
satisfy the algebraic equations for
where is the equilibrium current for fixed membrane potential V. The solutions of algebraic system (4) form a (1D) steady-state curve for model (1) with VC protocol (2), which is normally hyperbolic (transversally attracting) for
. For
the increase of
with speed
introduces a slow variation of all states
. Hence, V and the feedback current
(as measured), are not at their steady-state values given by (4), but they are changing dynamically. This results in a difference between the measured curve
in Fig 1B and the (I,V)-values of the desired steady-state I–V curve given by (4).
Geometrical singular perturbation theory (GSPT) by Fenichel [35] implies that after decay of initial transients every trajectory of the general model (1) with VC protocol (2) satisfies the algebraic conditions (4) for the steady-state curve up to order ε. The first-order terms in ε are
where is the time for deviations from the transversally stable steady-state I–V curve (4) to decay to half of their initial value; see eq (7) and Fig C in S1 Text for details. We estimate
from recovery transients after disturbances naturally occuring from imperfections in the voltage clamp during VC runs as
s (see Fig CB in S1 Text), such that
mV. Thus, the systematic bias between
in Fig 1B and the true steady state curve
caused by dynamically changing
is below measurement disturbances.
Deviation estimate (5) can also be tested in silico. Figs 3 and 4 emulate both VC and CC protocols with the Morris-Lecar model [33] (see Materials and methods, equation (10)), a biophysical model typically used for excitatory neurons of either class-I (Fig 3A) or class-II (Fig 4) excitability, and with the Wang-Buzsáki model (Fig 3B), which is a classical model of inhibitory interneuron [34] (see Materials and methods, equation (12)). Both models are of general form (1) with j = nj = 1. For the chosen parameter set (see S1 Text), the curves and
are order ε (
%) apart.
Consequently, time profile follows closely the steady-state I–V curve (4) of stable steady states of the fast subsystem (1) with VC protocol
, treating
as a parameter. This implies that faster ramp speeds are permissible when optimising trade-off between drift of cell properties and bias due to nonzero ramp speed. At high voltages the factor
becomes large, such that estimate (5) predicts larger deviations for large
, as confirmed in Figs 1B, 2, AB, AC and AD in S1 Text.
We now connect the steady-state I–V curve (4) of the VC protocol to a curve of fast-subsystem equilibria of the CC protocol, which is in part unstable. To this end we recast the VC protocol in the form of a CC protocol with non-constant current ramp speed and disturbances: the smoothed time profile
in Fig 1B of the VC run (thick, in blue/brown/red) equals the raw-data measured time profile
(thin blue curve with fluctuations) plus disturbances
, defined by
. After smoothing, the derivative
w.r.t.
is moderate (
pA/mV in modulus at its maximum near
), such that during the VC protocol the external current
satisfies
where . Hence,
, with upper bound
pA/ms in the range of Fig 1B. Thus,
is indeed still slow. So, except for disturbances
, the external current
is slowly varying according to a CC protocol with slowly time-varying speed
, such that the VC protocol (2) is equivalent to the CC protocol (6) with disturbances
.
This means that general model (1) with current given in (6) with zero disturbances (
) is a model for a CC protocol with driving current
, in contrast to the model with open-loop CC protocol (1), (3). Both models have the same fast subsystem (1) when setting
(i.e., for constant input current) and identifying
and
. The respective fast-subsystem steady states
satisfy
However, the two models differ by the nature of their respective slow variables and
:
is an externally applied hold current for the open-loop CC protocol (3), while
is a measured (and smoothed) current from the closed-loop feedback control
of the voltage source for (6). Thus, while the S-shaped steady-state curve
is identical for both models, it contains large unstable segments as a steady-state curve of open-loop CC protocol (1), (3), while it is always stable as a steady-state curve of closed-loop VC protocol (1), (6). The change in stability is caused by the disturbances
, which are current adjustments generated by the feedback term in VC protocol (2),
. Along most of the curve
the
are small fluctuations such that
and the feedback is approximately non-invasive [7]. Estimate (5) ensures that the measurements
stay close to
. Therefore, we can conclude that the VC protocol (2) with slowly varying feedback reference signal
reveals the entire family of steady states of a neuron (class I or II) with constant external current
, both stable (observable) and unstable (non-observable, hidden). Consequently, the N-shaped I-V relations for class-I neurons reported in [19,20,23] equal S-shaped steady-state bifurcation diagrams for these neurons with respect to
. They are tractable with a VC protocol where the current
is a sufficiently slowly varying feedback current with sufficiently small fluctuations
. In particular, this allows us to detect and pass through fold bifurcations directly in the experiment.
In contrast, during the CC open-loop protocol (3), when applying a slowly varying electrical current, the neuron dynamically transitions between its observable rest states and its observable (dynamically stable) spiking states (see Fig 1B orange time profile). The speed of variation pA/ms is such that
varies by 1 pA or less per spiking period. For a transient decay analysis we stimulate the neuron with larger current steps during a calibration phase before executing CC protocol runs. Fig 5A and 5B show the response to such a current step to
pA for cells 5 and 1. We observe that the transients in the step current response such as in Fig 5A and 5B in the stable spiking region have a half-time for decay toward the stable spikes
Panels A, B: time profiles of voltage responses from a current-clamp stimulation with a step to constant hold current
pA from 0.1 to 1.6 s (0.1–0.6 s in blue, 0.6–1.6 s in black). Red markings show how half-decay time
is extracted from voltage maxima during transients. Panels C, D:
phase-plane projection the time series from panels A, B, using a one-step finite-difference approximation of
with color code matching panels A, B to distinguish transients and steady-state spiking. Panels E, F: reproduction of Figs 1A and AA in S1 Text without bifurcation or stability markings, respectively. The value
pA (vertical dotted black line) and a 10 ms window around it (vertical solid black lines) are highlighted. The insets show a zoom into this 10 ms window around
pA of panels E, F. Black crosses in panels E, F are minimum and maximum of steady-state spiking from panels A, B for comparison.
This half-time enters estimates for the bias caused by varying the hold current
dynamically with speed
. For example, the estimate for the bias in the minimum voltage
of the spike equals to first order in ε
We can see in Fig 5E and 5F that changes in the spiking region with about 8 mV per 120 pA, so
mV
pA. The current changes with
pA/s. Thus, to first order in ε, the error then amounts to:
Hence, the effect from changing the current dynamically is small (below visibility in bifurcation diagrams such as Fig 1B or Fig 5E and 5F. The linear bias estimate grows to infinity when the amplitude
envelope assumes a square-root like shape and
goes to infinity, as is the case at Hopf bifurcations.
Thus, combining the VC protocol (6) for varying , and the CC protocol (3) for varying
, enables us to interpret the data sets from both protocols in Fig 1B as a bifurcation diagram including unstable states.
.
Drift and intermittent dropouts.
In Fig 5E and 5F the black markers (+) at pA mark the maximum and minimum values of the voltage response (after transient) to the step-current stimulation, as shown in panels A and B. They indicate that for PY cell 5 there is quantitative agreement between the response to the CC protocol with slowly-varying applied current
and the response to the current step. There is less agreement for PY cell 1, between step response and the response to the CC protocol with slow variation of
. The time profile suggests that there are either different possible spiking responses and, hence, more bifurcations in the dynamic protocol, such as period-doubling bifurcations, or the condition of the PY cell has drifted between step response calibration and CC ramp.
Furthermore, the distances between in the VC run and
in the CC run for the same cell are visibly larger than predicted by eq (5) along parts of the curve corresponding to dynamically stable stationary states. This is due to the natural drift of the neuron’s physiological properties as it changes conductance properties (e.g., temperature, degradation,...). In S1 Text we show that over time.
Finally, for the experimental curves presented in Figs 1B, 1C, 2 and A in S1 Text, the disturbances are not small in some unstable parts of the reported steady-state curve (e.g., near
mV in Fig 1B), caused by imperfect voltage clamping across the membrane. Fig CB in S1 Text shows a zoom demonstrating that the observed current spikes are indeed intermittent dropout events. Detailed simulations using Morris-Lecar model (10), shown in S1 Text Sect 3, reproduce these dropouts closely, if adds short current spikes and small-amplitude white noise to (10).
Discussion
Tracking non-observable (hidden) states and their stability boundaries in experimental settings bridges the gap between real-world phenomena and nonlinear science. Specifically, closed-loop control methods with slow variations of feedback reference signals enable to dissect the underlying states of multi-scale complex systems. Following this strategy, we demonstrate the feasibility of tracking a family of neuronal steady states (stable and unstable) via variations of reference signal and reparameterizing the obtained curve by feedback current. The method is versatile in that it applies to different cell types, both pyramidal (PY) neurons (like in Fig 1), of excitability class I or II (Figs 3A and 4), and inhibitory interneurons (Fig 3B).
Our analysis and the resulting estimate for the nonzero-speed induced bias during the VC protocol in (5) indicates that for the steady states detected in VC ramps our ramp speed (less than 3 mVs) does not create a bias that is notable when compared to drift of cell properties over time or fluctuations caused by external effects (such as the dropouts visible in Fig 1B). For the CC protocol and parameter areas were periodic spiking is stable, transient analysis for step current responses results in estimates such as (9), where bias caused by nonzero current ramp speed (
pA/s) is also negligible. This implies that faster ramp speeds are permissible when optimising trade-off between drift of cell properties and bias due to nonzero ramp speed. The single-ramp nature of the CC protocol leaves the question of stability open for some parts of the steady-state branch. These results can be improved by varying the hold current
non-monotonically and be switching from VC to CC protocol at selected parts of the steady-state branch determined by the VC protocol.
These results can be also extended to more complex neuronal states (e.g. limit cycles) by combining two recent advances, namely: 1. dynamic-clamp electrophysiology [2,36–40], which allows for a two-way real-time communication between a neuronal tissue (e.g. neuron, network or neuronal sub-processes) and a computer simulation of a component of the neuronal tissue. This introduces the full range of feedback control theory and engineering techniques into electrophysiology; 2. Control-based continuation for experiments (CBCE, which combines feedback control theory and pseudo-arclength numerical continuation for tracking solution branches of nonlinear systems directly from noisy experimental data [7]. Indeed, CBCE method has been successfully applied in various experimental systems, for instance, in mechanical vibration and buckling experiments [8], pedestrian flow experiments [9], cylindrical pipe flow simulations [11] and feasibility studies for synthetic gene networks [12,13]. Noteworthy, the methodology presented in this work can be seen as a special case of CBCE. Future work will focus on employing the full CBCE to track more complex neuronal states, for instance, unstable spiking states in single neurons or unstable traveling-wave states in neural tissue using micro-electrode arrays (MEA). This is crucial as it will allow us to validate computational models by comparing their numerical bifurcation diagrams with experimental ones; it will also help obtaining better model fitting based on both voltage-clamp and current-clamp data. This approach will have direct impact in experimental labs enabling biologists to have access to novel states with which to carry out new experimental paradigms. We envisage this extended protocol could be used to develop next-generation closed-loop deep-brain stimulation devices to treat certain brain diseases such as epilepsy [41], where it could help monitor control the excitability threshold of key neural populations in real time.
Materials and methods
Ethics statement
All animal treatments were authorized by the Sechenov Institute of Evolutionary Physiology and Biochemistry Bioethics Committee (protocol no. 1-7/2022, 27 January 2022) and adhered to the European Community Council Directive 86/609/EEC.
Protocols
Male Wistar rats were used in this study (age P21, N=8 animals). For the bifurcation diagrams (see Figs 1B and A in S1 Text), 4 of these 8 rats were used. PY Cells 1-3 were recorded from different animals, PY cells 4-5 were recorded from the same animal. To study the effects of hysteresis (see Fig 2) 20 neurons were recorded from the 4 remaining animals (221222, 230110, 230111, 230112). The control group were 6 neurons from 2 rats (230110, 230112). The QX group were 8 neurons from 4 rats (221222, 230110, 230111, 230112). The QX+Cd group were 6 neurons from 4 rats (221222, 230110, 230111, 230112).
The use and handling of animals was performed in accordance with the European Community Council Directive 86/609/EEC. Horizontal 300-μm-thick brain slices were prepared as described in our previous studies [42]. The slices contained the hippocampus and the adjacent cortical regions and were kept in the artificial cerebrospinal fluid (ACSF) with the following composition (in mM): 126 NaCl, 24 NaHCO3, 2.5 KCl, 2 CaCl2, 1.25 NaH2PO4, 1 MgSO4, 10 glucose (bubbled with 95% O2/5% CO2 gas mixture). All the listed chemicals were purchased from Sigma-Aldrich (St. Louis, MO, USA).
We performed the whole-cell patch-clamp recordings of the principal neurons in the entorhinal cortex. Neurons within the slices were visualized using a Zeiss Axioscop 2 microscope (Zeiss, Oberkochen, Germany), equipped with a digital camera (Grasshopper 3 GS3-U3-23S6M-C; FLIR Integrated Imaging Solutions Inc., Wilsonville, OR, USA) and differential interference contrast optics.
Patch pipettes were produced from borosilicate glass capillaries (Sutter Instrument, Novato, CA, USA) and filled with one of the following pipette solutions. A potassium gluconate-based pipette solution had the following composition (in mM): 136-K-Gluconate, 10 NaCl, 10 HEPES, 5 EGTA, 4 ATP-Mg, 0.3 GTP. For control experiments described in Section SI4, the intracellular block of the voltage-gated sodium channels was required, for which we used the solution with added QX314 (Alamone labs, Jerusalem, Israel), which had the following composition (in mM): 130-K-Gluconate, 10 HEPES, 6 QX314, 6 KCl, 5 EGTA, 4 ATP-Mg, 2 NaCl, 0.3 GTP. The pH of both solutions was adjusted to 7.25 with KOH. The resistance of filled patch-pipettes was 3-4 MΩ.
A Multiclamp 700B (Molecular Devices, Sunnyvale, CA, USA) patch-clamp amplifier, a NI USB-6343 A/D converter (National Instruments, Austin, TX, USA) and WinWCP 5 software (SIPBS, UK) were used to obtain the electrophysiological data. The recordings were lowpass filtered at 10 kHz and sampled at 20-30 kHz. The access resistance was less than 15 MΩ and remained stable during the recordings. The liquid junction potential was not compensated for. The flow rate of ACSF in the recording chamber was 5 ml/min. The recordings were performed at 30°C.
Specifically for the voltage-clamp protocol. We note the CV-7B headstage has four different feedback resistors (Rf): 50 MΩ, 500 MΩ, 5 GΩ, and 50 GΩ. The Rf determines the maximum currents that can be recorded or injected. In voltage-clamp mode it is generally recommended to use the largest possible value of Rf (larger Rf results in less noise), though high values can result in current saturation. Since in our preparation the electrical currents varied between 50-2000 pA (several nA for the potassium ion currents at positive holding potentials), we chose Rf = 500 MΩ (i.e. feedback gain gc = 2 nS).
We applied first VC and then CC protocol to 5 neurons in the entorhinal cortex of 4 male Wistar rats [42]. We first performed the VC neuronal recordings varying the hold voltage from –80 mV to + 30 mV slowly with
mV/s, while measuring current, called
in Figs 1B and 5C. Subsequently, for the CC recordings we first determine the minimal injected current required to induce a depolarization block of action potential generation (
in Fig 1B, upper limit of current input where firing occurs). Then we injected hold current
, gradually increasing from 0 pA to
during 60 s, such that
is in the range
pA/s for the 5 neurons, while recording voltage
. See S1 Text for further checks (e.g., for hysteresis).
Computational models
We applied the VC and the CC protocols, with slow variations in the feedback reference signal and in the applied current, respectively, to the classical neuron model adapted to many contexts, namely the Morris-Lecar model [33] (ML), and to a standard neuron model that was specifically designed to account for inhibitory neural activity, namely the Wang-Buzsáki model [34] (WB). Parameter values are given in Tables 1 and 2 below, respectively.
The ML equations are as follows
with the following voltage-dependent (in)activation and time-constant functions:
To obtain Figs 3A and 4, we have used the following parameter values.
Finally, for both the VC protocol with slow variation, and the CC protocol with slow variation, the speed of the variation was chosen to be equal to . Note that gc can be decreased to 7, which is in the same order of magnitude as the experimental one. Moreover, the capacitance chosen for the model simulations is on the same order of magnitude as observed in the experiments. Nevertheless, the key point to note is that we are not aiming for quantitative agreement since this is a conductance-based phenomenological model.
The WB equations are as follows
with the following voltage-dependent (in)activation and time-constant functions:
To obtain Fig 3B, we have used the following parameter values.
Finally, for both the VC protocol with slow variation, and the CC protocol with slow variation, the speed of the variation was chosen to be equal to .
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