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Observing hidden neuronal states in experiments

Dmitry Amakhin, Anton Chizhov, Guillaume Girier, Mathieu Desroches, Jan Sieber, Serafim Rodrigues

TL;DR

This work presents a model-independent experimental framework to map both stable and unstable neuronal steady states by coupling a slowly ramped voltage-clamp protocol with a corresponding current-clamp protocol on the same neuron. Through a multiple-timescale analysis, the voltage clamp acts as a slow feedback that reveals the fast subsystem's steady states, yielding an experimental I–V curve that includes unstable branches when overlaid with CC data. The authors derive a first-order ramp-bias estimate and establish conditions (spectral gap and observability) under which the measured VC curve accurately reflects the underlying steady states, enabling direct observation of bifurcations such as folds and Hopf points. They demonstrate the approach across entorhinal PY cells and interneurons, examine drift and intermittent fluctuations, and validate the method with in-silico models, outlining a path toward richer closed-loop neurophysiological investigations and potential clinical applications in neuromodulation.

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.

Observing hidden neuronal states in experiments

TL;DR

This work presents a model-independent experimental framework to map both stable and unstable neuronal steady states by coupling a slowly ramped voltage-clamp protocol with a corresponding current-clamp protocol on the same neuron. Through a multiple-timescale analysis, the voltage clamp acts as a slow feedback that reveals the fast subsystem's steady states, yielding an experimental I–V curve that includes unstable branches when overlaid with CC data. The authors derive a first-order ramp-bias estimate and establish conditions (spectral gap and observability) under which the measured VC curve accurately reflects the underlying steady states, enabling direct observation of bifurcations such as folds and Hopf points. They demonstrate the approach across entorhinal PY cells and interneurons, examine drift and intermittent fluctuations, and validate the method with in-silico models, outlining a path toward richer closed-loop neurophysiological investigations and potential clinical applications in neuromodulation.

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.
Paper Structure (20 sections, 36 equations, 14 figures, 2 tables)

This paper contains 20 sections, 36 equations, 14 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Results
  3. Analysis
  4. Drift and intermittent dropouts
  5. Discussion
  6. Materials and Methods
  7. Ethics Statement
  8. Protocols
  9. Computational models
  10. Experimental bifurcation diagrams for Pyramidal (PY) cells 1--5, including current and voltage clamp ramp data
  11. Spiking frequency and interspike intervals during CC protocol
  12. Details of intermittent fluctuations around unstable steady states
  13. Experiments estimating hysteresis effects and blocking of voltage-gated sodium and calcium ion channels
  14. Equilibrium analysis of the general conductance-based model for the $\mathbf{V}$-clamped protocol
  15. Conditions for validity of estimate for systematic bias due to dynamic ramp of voltage
  16. ...and 5 more sections

Figures (14)

  • Figure 1: A: Sketch of experimental setup with brain slice (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 $5$: $(I_\mathrm{vc}(t),V_{\mathrm{h}}(t))$-curve for VC run (thin bright blue: unfiltered data with sampling time step $5\times10^{-5}$ s, blue/red/brown: median of $I_\mathrm{vc}$ over moving windows of size $\Delta w=4\times10^{3}$ steps equalling $0.2$s) and $(I_{\mathrm{h}}(t),V_\mathrm{cc}(t))$-curve for CC run (orange, mean of $I_{\mathrm{h}}$ over moving windows. C: $(I_\mathrm{vc}(t),V_{\mathrm{h}}(t))$-curves for VC protocol of all $5$ cells on waterfall $V_{\mathrm{h}}$-axis (color coding indicates conjectured stability as indicated for (b)).
  • Figure 2: A: Experimental bifurcation diagram for an interneuron from the entorhinal cortex. B: Experimental bifurcation diagram for a class-II PY neuron from the same region. Protocols identical and color coding to Fig \ref{['fig:fig1']}B.
  • Figure 3: VC and CC in silico. Protocol as described for Fig \ref{['fig:fig1']}B applied to A: a class-I Morris-Lecar neuron model \ref{['eq:ML']}morris1981 as example of excitatory cell, and B: a class-I Wang-Buzsáki neuron model \ref{['eq:WB']}wang1996 as example of inhibitory interneuron; see \ref{['eq:ML']} and \ref{['eq:WB']} for differential equations and Tables \ref{['tab:variables']} and \ref{['tab:variablesWB']} for parameter values. The two-dimensional fast subsystem has a S-shaped steady-state curve satisfying steady-state conditions \ref{['eq:VCSRfast1eq']}. The steady-state $I$--$V$ curve \ref{['eq:VCSRfast1eq']} and the (multi-color) S-shaped curve from the VC protocol \ref{['eq:VCSR']} are indistinguishable throughout the range of input currents $I_\mathrm{vc}$. The orange curve resulting from the CC protocol is very close to $S^0$ and the VC protocol near its dynamically stable parts. See Figs F and G in S1 Text for numerical bifurcation diagrams.
  • Figure 4: VC and CC protocols as described for Fig \ref{['fig:fig1']}B applied to the Morris-Lecar model in a parameter regime where it behaves as a class-II neuron model. See \ref{['eq:ML']} for differential equations and Table \ref{['tab:variables']} for parameter values and Fig H in S1 Text for numerical bifurcation diagram.
  • Figure 5: Periodic spiking responses to step-current protocols for PY cells 5 (left column) and 1 (right column). Panels A, B: time profiles of voltage responses $V_\mathrm{cc}(t)$ from a current-clamp stimulation with a step to constant hold current $I_{\mathrm{h}}=200$ 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 $\tau_{1/2}$ is extracted from voltage maxima during transients. Panels C, D: $(V_\mathrm{cc},V_\mathrm{cc}')$ phase-plane projection the time series from panels A, B, using a one-step finite-difference approximation of $V_\mathrm{cc}^{\prime}(t)$ with color code matching panels A, B to distinguish transients and steady-state spiking. Panels E, F: reproduction of Figs \ref{['fig:fig1']}A and AA in S1 Text without bifurcation or stability markings, respectively. The value $I_{\mathrm{h}}=200$ 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 $I_{\mathrm{h}}=200$ 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.
  • ...and 9 more figures