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Biophysical basis for three distinct dynamical mechanisms of action potential initiation.

Prescott SA, De Koninck Y, Sejnowski TJ - PLoS Comput. Biol. (2008)

Bottom Line: Hodgkin identified three classes of neurons with qualitatively different analog-to-digital transduction properties.From this, we conclude that the spike-initiating dynamics associated with each of Hodgkin's classes represent different outcomes in a nonlinear competition between oppositely directed, kinetically mismatched currents.Through detailed analysis of the spike-initiating process, we have explained a fundamental link between biophysical properties and qualitative differences in how neurons encode sensory input.

View Article: PubMed Central - PubMed

Affiliation: Computational Neurobiology Laboratory, Salk Institute, La Jolla, California, United States of America. prescott@neurobio.pitt.edu

ABSTRACT
Transduction of graded synaptic input into trains of all-or-none action potentials (spikes) is a crucial step in neural coding. Hodgkin identified three classes of neurons with qualitatively different analog-to-digital transduction properties. Despite widespread use of this classification scheme, a generalizable explanation of its biophysical basis has not been described. We recorded from spinal sensory neurons representing each class and reproduced their transduction properties in a minimal model. With phase plane and bifurcation analysis, each class of excitability was shown to derive from distinct spike initiating dynamics. Excitability could be converted between all three classes by varying single parameters; moreover, several parameters, when varied one at a time, had functionally equivalent effects on excitability. From this, we conclude that the spike-initiating dynamics associated with each of Hodgkin's classes represent different outcomes in a nonlinear competition between oppositely directed, kinetically mismatched currents. Class 1 excitability occurs through a saddle node on invariant circle bifurcation when net current at perithreshold potentials is inward (depolarizing) at steady state. Class 2 excitability occurs through a Hopf bifurcation when, despite net current being outward (hyperpolarizing) at steady state, spike initiation occurs because inward current activates faster than outward current. Class 3 excitability occurs through a quasi-separatrix crossing when fast-activating inward current overpowers slow-activating outward current during a stimulus transient, although slow-activating outward current dominates during constant stimulation. Experiments confirmed that different classes of spinal lamina I neurons express the subthreshold currents predicted by our simulations and, further, that those currents are necessary for the excitability in each cell class. Thus, our results demonstrate that all three classes of excitability arise from a continuum in the direction and magnitude of subthreshold currents. Through detailed analysis of the spike-initiating process, we have explained a fundamental link between biophysical properties and qualitative differences in how neurons encode sensory input.

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Related in: MedlinePlus

Competition between kinetically mismatched currents.(A) Top panels show individual currents in 2D model; bottom panels show                            how they combine to produce the instantaneous                                (Iinst) and steady state                                (Iss) I–V                            curves. Double-headed arrows highlight effect of                                βw on the voltage-dependency of                                Islow. Class 3 neuron:                                Islow activates at lower                            V than Ifast, meaning slow                            negative feedback keeps V from increasing high enough                            to initiate fast positive feedback at steady state.                            Fast positive feedback (that results in a spike) can be initiated only                            if the system is perturbed from steady state. Quasi-separatrix (blue)                            has a region of negative slope (*) indicating where net positive                            feedback occurs given the kinetic difference between fast and slow                            currents: positive feedback that activates rapidly can compete                            effectively with stronger negative feedback whose full activation is                            delayed by its slower kinetics. If V is forced rapidly                            past the blue arrowhead, fast positive feedback initiates a single spike                            before slow negative feedback catches up and forces the system back to                            its stable fixed point. Quasi-separatrix is plotted as the sum of all                            currents but with Islow calculated as a                            function of w at the quasi-separatrix (see phase plane                            in Figure 2A) rather                            than at steady state and is shown here for                            Istim = 60                                µA/cm2. Class 2 neuron:                                Islow and                            Ifast activate at roughly the same                            V. A Hopf bifurcation occurs at the point indicated by                            the arrow, where  (see Results).                            This means that fast positive feedback exceeds slow negative feedback at                            steady state; as for class 3 neurons, this relies on positive feedback                            having fast kinetics since the net perithreshold current is still                            outward (i.e., steady state I–V curve is                            monotonic). Note that the slope of the steady-state                                I–V curve is less steep in the class 2                            model than in the class 3 model. Class 1 neuron:                                Islow activates at higher                            V than Ifast, meaning slow                            negative feedback does not begin activating until after the spike is                            initiated. This gives a steady state                                I–V curve that is                            non-monotonic with a region of negative slope (*) near the apex                            of the instantaneous I–V                            curve. The SNIC bifurcation occurs when                                ∂Iss/∂t = 0                            (arrowhead) because, at this voltage, Ifast                            counterbalances Ileak and any further                            depolarization will cause progressive activation of                                Ifast. (B) Changing                                ḡfast in the 2D model had                            equivalent effects on the shape of the steady state                                I–V curves. Unlike in (A), voltage at the                            apex of the instantaneous                            I–V curve (purple arrows)                            changes as ḡfast is varied; in                            other words, the net current at perithreshold potentials can be                            modulated by changing fast currents (which directly impact voltage                            threshold) rather than by changing the amplitude or voltage-dependency                            of slow currents. This is consistent with results in Figure 8. (C) Speeding                            up the kinetics of Islow impacts the onset                            of class 2 and 3 excitability. Compared with original model                                (φw = 0.15;                            black), increasing φw to 0.25                            (red) increased Istim required to cause a                            Hopf bifurcation or a QSC, but did not affect                                Istim required to cause an SNIC                            bifurcation; reducing φw to 0.10                            (green) had the opposite effect (summarized in right panel). Increasing                                φw also widened the                            discontinuity in the class 2 f–I curve and                            allowed class 2 and 3 neurons to achieve higher spiking rates with                            strong Istim because of the faster recovery                            between spikes; reducing φw had                            the opposite effects.
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pcbi-1000198-g009: Competition between kinetically mismatched currents.(A) Top panels show individual currents in 2D model; bottom panels show how they combine to produce the instantaneous (Iinst) and steady state (Iss) I–V curves. Double-headed arrows highlight effect of βw on the voltage-dependency of Islow. Class 3 neuron: Islow activates at lower V than Ifast, meaning slow negative feedback keeps V from increasing high enough to initiate fast positive feedback at steady state. Fast positive feedback (that results in a spike) can be initiated only if the system is perturbed from steady state. Quasi-separatrix (blue) has a region of negative slope (*) indicating where net positive feedback occurs given the kinetic difference between fast and slow currents: positive feedback that activates rapidly can compete effectively with stronger negative feedback whose full activation is delayed by its slower kinetics. If V is forced rapidly past the blue arrowhead, fast positive feedback initiates a single spike before slow negative feedback catches up and forces the system back to its stable fixed point. Quasi-separatrix is plotted as the sum of all currents but with Islow calculated as a function of w at the quasi-separatrix (see phase plane in Figure 2A) rather than at steady state and is shown here for Istim = 60 µA/cm2. Class 2 neuron: Islow and Ifast activate at roughly the same V. A Hopf bifurcation occurs at the point indicated by the arrow, where (see Results). This means that fast positive feedback exceeds slow negative feedback at steady state; as for class 3 neurons, this relies on positive feedback having fast kinetics since the net perithreshold current is still outward (i.e., steady state I–V curve is monotonic). Note that the slope of the steady-state I–V curve is less steep in the class 2 model than in the class 3 model. Class 1 neuron: Islow activates at higher V than Ifast, meaning slow negative feedback does not begin activating until after the spike is initiated. This gives a steady state I–V curve that is non-monotonic with a region of negative slope (*) near the apex of the instantaneous I–V curve. The SNIC bifurcation occurs when ∂Iss/∂t = 0 (arrowhead) because, at this voltage, Ifast counterbalances Ileak and any further depolarization will cause progressive activation of Ifast. (B) Changing ḡfast in the 2D model had equivalent effects on the shape of the steady state I–V curves. Unlike in (A), voltage at the apex of the instantaneous I–V curve (purple arrows) changes as ḡfast is varied; in other words, the net current at perithreshold potentials can be modulated by changing fast currents (which directly impact voltage threshold) rather than by changing the amplitude or voltage-dependency of slow currents. This is consistent with results in Figure 8. (C) Speeding up the kinetics of Islow impacts the onset of class 2 and 3 excitability. Compared with original model (φw = 0.15; black), increasing φw to 0.25 (red) increased Istim required to cause a Hopf bifurcation or a QSC, but did not affect Istim required to cause an SNIC bifurcation; reducing φw to 0.10 (green) had the opposite effect (summarized in right panel). Increasing φw also widened the discontinuity in the class 2 f–I curve and allowed class 2 and 3 neurons to achieve higher spiking rates with strong Istim because of the faster recovery between spikes; reducing φw had the opposite effects.

Mentions: Interpretation of the phase plane geometry can be formalized by doing local stability analysis near the fixed points ([27], see also chapter 11 in [28]). In class 3 neurons, at the stable fixed point. This means, at steady state, that positive feedback is slower than the rate of negative feedback, φw/τw. Subthreshold activation of Islow produces a steady state I–V curve that is monotonic and sufficiently steep near the apex of the instantaneous I–V curve that V is prohibited from rising high enough to strongly activate Ifast (Figure 9A, left). However, because the two feedback processes have different kinetics, a spike can be initiated if the system is perturbed from steady state: if V escapes high enough to activate Ifast (e.g., at the onset of an abrupt step in Istim), fast-activating inward current can overpower slow-activating outward current—the latter is stronger when fully activated, but can only partially activate (because of its slow kinetics) before a spike is inevitable. Through this mechanism, a single spike can be initiated before negative feedback forces the system back to its stable fixed point, hence class 3 excitability. Speeding up the kinetics of Islow predictably allows Islow to compete more effectively with Ifast (see below).


Biophysical basis for three distinct dynamical mechanisms of action potential initiation.

Prescott SA, De Koninck Y, Sejnowski TJ - PLoS Comput. Biol. (2008)

Competition between kinetically mismatched currents.(A) Top panels show individual currents in 2D model; bottom panels show                            how they combine to produce the instantaneous                                (Iinst) and steady state                                (Iss) I–V                            curves. Double-headed arrows highlight effect of                                βw on the voltage-dependency of                                Islow. Class 3 neuron:                                Islow activates at lower                            V than Ifast, meaning slow                            negative feedback keeps V from increasing high enough                            to initiate fast positive feedback at steady state.                            Fast positive feedback (that results in a spike) can be initiated only                            if the system is perturbed from steady state. Quasi-separatrix (blue)                            has a region of negative slope (*) indicating where net positive                            feedback occurs given the kinetic difference between fast and slow                            currents: positive feedback that activates rapidly can compete                            effectively with stronger negative feedback whose full activation is                            delayed by its slower kinetics. If V is forced rapidly                            past the blue arrowhead, fast positive feedback initiates a single spike                            before slow negative feedback catches up and forces the system back to                            its stable fixed point. Quasi-separatrix is plotted as the sum of all                            currents but with Islow calculated as a                            function of w at the quasi-separatrix (see phase plane                            in Figure 2A) rather                            than at steady state and is shown here for                            Istim = 60                                µA/cm2. Class 2 neuron:                                Islow and                            Ifast activate at roughly the same                            V. A Hopf bifurcation occurs at the point indicated by                            the arrow, where  (see Results).                            This means that fast positive feedback exceeds slow negative feedback at                            steady state; as for class 3 neurons, this relies on positive feedback                            having fast kinetics since the net perithreshold current is still                            outward (i.e., steady state I–V curve is                            monotonic). Note that the slope of the steady-state                                I–V curve is less steep in the class 2                            model than in the class 3 model. Class 1 neuron:                                Islow activates at higher                            V than Ifast, meaning slow                            negative feedback does not begin activating until after the spike is                            initiated. This gives a steady state                                I–V curve that is                            non-monotonic with a region of negative slope (*) near the apex                            of the instantaneous I–V                            curve. The SNIC bifurcation occurs when                                ∂Iss/∂t = 0                            (arrowhead) because, at this voltage, Ifast                            counterbalances Ileak and any further                            depolarization will cause progressive activation of                                Ifast. (B) Changing                                ḡfast in the 2D model had                            equivalent effects on the shape of the steady state                                I–V curves. Unlike in (A), voltage at the                            apex of the instantaneous                            I–V curve (purple arrows)                            changes as ḡfast is varied; in                            other words, the net current at perithreshold potentials can be                            modulated by changing fast currents (which directly impact voltage                            threshold) rather than by changing the amplitude or voltage-dependency                            of slow currents. This is consistent with results in Figure 8. (C) Speeding                            up the kinetics of Islow impacts the onset                            of class 2 and 3 excitability. Compared with original model                                (φw = 0.15;                            black), increasing φw to 0.25                            (red) increased Istim required to cause a                            Hopf bifurcation or a QSC, but did not affect                                Istim required to cause an SNIC                            bifurcation; reducing φw to 0.10                            (green) had the opposite effect (summarized in right panel). Increasing                                φw also widened the                            discontinuity in the class 2 f–I curve and                            allowed class 2 and 3 neurons to achieve higher spiking rates with                            strong Istim because of the faster recovery                            between spikes; reducing φw had                            the opposite effects.
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pcbi-1000198-g009: Competition between kinetically mismatched currents.(A) Top panels show individual currents in 2D model; bottom panels show how they combine to produce the instantaneous (Iinst) and steady state (Iss) I–V curves. Double-headed arrows highlight effect of βw on the voltage-dependency of Islow. Class 3 neuron: Islow activates at lower V than Ifast, meaning slow negative feedback keeps V from increasing high enough to initiate fast positive feedback at steady state. Fast positive feedback (that results in a spike) can be initiated only if the system is perturbed from steady state. Quasi-separatrix (blue) has a region of negative slope (*) indicating where net positive feedback occurs given the kinetic difference between fast and slow currents: positive feedback that activates rapidly can compete effectively with stronger negative feedback whose full activation is delayed by its slower kinetics. If V is forced rapidly past the blue arrowhead, fast positive feedback initiates a single spike before slow negative feedback catches up and forces the system back to its stable fixed point. Quasi-separatrix is plotted as the sum of all currents but with Islow calculated as a function of w at the quasi-separatrix (see phase plane in Figure 2A) rather than at steady state and is shown here for Istim = 60 µA/cm2. Class 2 neuron: Islow and Ifast activate at roughly the same V. A Hopf bifurcation occurs at the point indicated by the arrow, where (see Results). This means that fast positive feedback exceeds slow negative feedback at steady state; as for class 3 neurons, this relies on positive feedback having fast kinetics since the net perithreshold current is still outward (i.e., steady state I–V curve is monotonic). Note that the slope of the steady-state I–V curve is less steep in the class 2 model than in the class 3 model. Class 1 neuron: Islow activates at higher V than Ifast, meaning slow negative feedback does not begin activating until after the spike is initiated. This gives a steady state I–V curve that is non-monotonic with a region of negative slope (*) near the apex of the instantaneous I–V curve. The SNIC bifurcation occurs when ∂Iss/∂t = 0 (arrowhead) because, at this voltage, Ifast counterbalances Ileak and any further depolarization will cause progressive activation of Ifast. (B) Changing ḡfast in the 2D model had equivalent effects on the shape of the steady state I–V curves. Unlike in (A), voltage at the apex of the instantaneous I–V curve (purple arrows) changes as ḡfast is varied; in other words, the net current at perithreshold potentials can be modulated by changing fast currents (which directly impact voltage threshold) rather than by changing the amplitude or voltage-dependency of slow currents. This is consistent with results in Figure 8. (C) Speeding up the kinetics of Islow impacts the onset of class 2 and 3 excitability. Compared with original model (φw = 0.15; black), increasing φw to 0.25 (red) increased Istim required to cause a Hopf bifurcation or a QSC, but did not affect Istim required to cause an SNIC bifurcation; reducing φw to 0.10 (green) had the opposite effect (summarized in right panel). Increasing φw also widened the discontinuity in the class 2 f–I curve and allowed class 2 and 3 neurons to achieve higher spiking rates with strong Istim because of the faster recovery between spikes; reducing φw had the opposite effects.
Mentions: Interpretation of the phase plane geometry can be formalized by doing local stability analysis near the fixed points ([27], see also chapter 11 in [28]). In class 3 neurons, at the stable fixed point. This means, at steady state, that positive feedback is slower than the rate of negative feedback, φw/τw. Subthreshold activation of Islow produces a steady state I–V curve that is monotonic and sufficiently steep near the apex of the instantaneous I–V curve that V is prohibited from rising high enough to strongly activate Ifast (Figure 9A, left). However, because the two feedback processes have different kinetics, a spike can be initiated if the system is perturbed from steady state: if V escapes high enough to activate Ifast (e.g., at the onset of an abrupt step in Istim), fast-activating inward current can overpower slow-activating outward current—the latter is stronger when fully activated, but can only partially activate (because of its slow kinetics) before a spike is inevitable. Through this mechanism, a single spike can be initiated before negative feedback forces the system back to its stable fixed point, hence class 3 excitability. Speeding up the kinetics of Islow predictably allows Islow to compete more effectively with Ifast (see below).

Bottom Line: Hodgkin identified three classes of neurons with qualitatively different analog-to-digital transduction properties.From this, we conclude that the spike-initiating dynamics associated with each of Hodgkin's classes represent different outcomes in a nonlinear competition between oppositely directed, kinetically mismatched currents.Through detailed analysis of the spike-initiating process, we have explained a fundamental link between biophysical properties and qualitative differences in how neurons encode sensory input.

View Article: PubMed Central - PubMed

Affiliation: Computational Neurobiology Laboratory, Salk Institute, La Jolla, California, United States of America. prescott@neurobio.pitt.edu

ABSTRACT
Transduction of graded synaptic input into trains of all-or-none action potentials (spikes) is a crucial step in neural coding. Hodgkin identified three classes of neurons with qualitatively different analog-to-digital transduction properties. Despite widespread use of this classification scheme, a generalizable explanation of its biophysical basis has not been described. We recorded from spinal sensory neurons representing each class and reproduced their transduction properties in a minimal model. With phase plane and bifurcation analysis, each class of excitability was shown to derive from distinct spike initiating dynamics. Excitability could be converted between all three classes by varying single parameters; moreover, several parameters, when varied one at a time, had functionally equivalent effects on excitability. From this, we conclude that the spike-initiating dynamics associated with each of Hodgkin's classes represent different outcomes in a nonlinear competition between oppositely directed, kinetically mismatched currents. Class 1 excitability occurs through a saddle node on invariant circle bifurcation when net current at perithreshold potentials is inward (depolarizing) at steady state. Class 2 excitability occurs through a Hopf bifurcation when, despite net current being outward (hyperpolarizing) at steady state, spike initiation occurs because inward current activates faster than outward current. Class 3 excitability occurs through a quasi-separatrix crossing when fast-activating inward current overpowers slow-activating outward current during a stimulus transient, although slow-activating outward current dominates during constant stimulation. Experiments confirmed that different classes of spinal lamina I neurons express the subthreshold currents predicted by our simulations and, further, that those currents are necessary for the excitability in each cell class. Thus, our results demonstrate that all three classes of excitability arise from a continuum in the direction and magnitude of subthreshold currents. Through detailed analysis of the spike-initiating process, we have explained a fundamental link between biophysical properties and qualitative differences in how neurons encode sensory input.

Show MeSH
Related in: MedlinePlus