Gain control with A-type potassium current: IA as a switch between divisive and subtractive inhibition

Neurons process information by transforming barrages of synaptic inputs into spiking activity. Synaptic inhibition suppresses the output firing activity of a neuron, and is commonly classified as having a subtractive or divisive effect on a neuron's output firing activity. Subtractive inhibition can narrow the range of inputs that evoke spiking activity by eliminating responses to non-preferred inputs. Divisive inhibition is a form of gain control: it modifies firing rates while preserving the range of inputs that evoke firing activity. Since these two "modes" of inhibition have distinct impacts on neural coding, it is important to understand the biophysical mechanisms that distinguish these response profiles. We use simulations and mathematical analysis of a neuron model to find the specific conditions for which inhibitory inputs have subtractive or divisive effects. We identify a novel role for the A-type Potassium current (IA). In our model, this fast-activating, slowly- inactivating outward current acts as a switch between subtractive and divisive inhibition. If IA is strong (large maximal conductance) and fast (activates on a time-scale similar to spike initiation), then inhibition has a subtractive effect on neural firing. In contrast, if IA is weak or insufficiently fast-activating, then inhibition has a divisive effect on neural firing. We explain these findings using dynamical systems methods to define how a spike threshold condition depends on synaptic inputs and IA. Our findings suggest that neurons can "self-regulate" the gain control effects of inhibition via combinations of synaptic plasticity and/or modulation of the conductance and kinetics of A-type Potassium channels. This novel role for IA would add flexibility to neurons and networks, and may relate to recent observations of divisive inhibitory effects on neurons in the nucleus of the solitary tract.Comment: 20 pages, 11 figure

Inhibition is subtractive for large A-channel conductance or weak synaptic excitation.

<p><b>A, B</b>: Firing rates computed from simulations with inhibition (<i>g</i>_{<i>Syn</i>,<i>I</i>} = 1, <i>r</i>_<i>I</i> = 50 Hz, abscissa) plotted as a function of firing rates computed from simulations without inhibition (<i>g</i>_{<i>Syn</i>,<i>I</i>} = 0, ordinate). In <b>A</b>: Three values of A-channel conductance are compared (<i>g</i>_<i>A</i> = 20, 30, 40) with synaptic excitation strength fixed at <i>g</i>_{<i>Syn</i>,<i>E</i>} = 0.5. Inhibition is subtractive for large <i>g</i>_<i>A</i> evident in the rightward shift of the threshold-linear relationship between firing rates for <i>g</i>_<i>A</i> = 40. In <b>B</b>: Three values of synaptic excitation strength are compared (<i>g</i>_{<i>Syn</i>,<i>E</i>} = 0.4, 0.5, 0.7) with A-channel conductance fixed at <i>g</i>_<i>A</i> = 30. Inhibition is subtractive for weaker excitation, evident in the rightward shift of the threshold-linear relationship between firing rates for <i>g</i>_{<i>Syn</i>,<i>E</i>} = 0.4.</p

Dependence of the <i>V</i>-nullcline on A: <i>g</i>_<i>A</i>, B: <i>b</i>, C: <i>s</i>_<i>I</i> and D: <i>s</i>_<i>E</i>.

<p>Default values of the parameters are <i>g</i>_<i>A</i> = 20, <i>b</i> = .5, <i>s</i>_<i>I</i> = .5 and <i>s</i>_<i>E</i> = 1. Moreover, <i>g</i>_{<i>Syn</i>,<i>E</i>} = 3 and <i>g</i>_{<i>Syn</i>,<i>I</i>} = 5. Thin blue line is <i>n</i>_∞(<i>V</i>), the <i>n</i>-nullcline.</p

Response to an excitatory input.

<p><b>A</b>. The neuron will or will not fire an action potential if, at the time of the excitatory input, it lies below or above the left knee of the <i>s</i>_<i>E</i> = 1 cubic, respectively. <b>B</b>. The neuron cannot respond with an action potential if the left knee of the <i>s</i>_<i>E</i> = 1 cubic lies below the <i>n</i> = 0 axis.</p

Comparison of firing rate input/output relations for subtractive and divisive inhibition (illustration only, not actual data).

<p><b>A</b>: Subtractive inhibition: output rate without inhibition is , and output rate with inhibition is , where <i>c</i> is a constant with <i>c</i> > 0. <b>B</b>: Divisive inhibition: output rate without inhibition is (same as in <b>A</b>), and output rate with with inhibition is , where <i>α</i> is a constant with 0 < <i>α</i> < 1.</p

Divisive and subtractive inhibition in a multi-compartment neuron model.

<p><b>A</b>: Voltage traces in response to excitatory inputs at varying input locations along the dendrite. Parameter values in these simulations: <i>g</i>_{<i>Syn</i>,<i>E</i>} = 3, <i>g</i>_{<i>Syn</i>,<i>I</i>} = 0, and <i>g</i>_<i>A</i> = 0. Inputs distant from the soma lead to spike initiation with millisecond-scale delay between excitatory input and spike onset. <b>B</b>: Threshold-linear relation between output firing rates in simulations of the multi-compartment model with and without inhibition for varying input location and <i>g</i>_<i>A</i> = 20. For simulations with inhibition: <i>g</i>_{<i>Syn</i>,<i>I</i>} = 1 and <i>r</i>_<i>I</i> = 50. Inhibition is subtractive for distal excitatory input (<i>cpt</i>_<i>in</i> = 6). <b>C</b>: Critical values of <i>g</i>_<i>A</i> that define boundary between subtractive and divisive inhibition in (<i>g</i>_{<i>Syn</i>,<i>E</i>}, <i>g</i>_<i>A</i>) parameter space. The boundary shifts downward as excitatory inputs are moved to more distal locations, indicating that inhibition has a subtractive effect for lower values of <i>g</i>_<i>A</i> for more distal inputs.</p

Boundary between subtractive and divisive inhibition in (<i>g</i>_{<i>Syn</i>,<i>E</i>}, <i>g</i>_<i>A</i>) parameter space.

<p><b>A, B</b>: For each parameter set, we fit threshold-linear functions to characterize the relationship between output firing rates in the presence and absence of inhibition. Dots in each panel identify the smallest value of <i>g</i>_<i>A</i> (for a given parameter set) at which inhibition is subtractive. In <b>A</b>: We vary inhibition strength (<i>g</i>_{<i>Syn</i>,<i>I</i>} = 0.5, 1, 2) and keep inhibition rate fixed at 50 Hz. In <b>B</b>: We vary inhibition rate (<i>r</i>_<i>I</i> = 30, 50, 70 Hz) and keep inhibition strength fixed at <i>g</i>_{<i>Syn</i>,<i>I</i>} = 1. The values of <i>g</i>_<i>A</i> that define the boundary between subtractive and divisive inhibition decrease with increases in either inhibition parameter (<i>g</i>_{<i>Syn</i>,<i>I</i>} or <i>r</i>_<i>I</i>).</p

Examples of divisive and subtractive effects of inhibition in the one-compartment model.

<p><b>A, B</b>: Output firing rates as a function of excitatory input rate, computed from simulations without inhibition (empty circles, <i>g</i>_{<i>Syn</i>,<i>I</i>} = 0) and with inhibition (filled circles, <i>g</i>_{<i>Syn</i>,<i>I</i>} = 1 and <i>r</i>_<i>I</i> = 50 Hz). Excitatory synaptic strength is <i>g</i>_{<i>Syn</i>,<i>E</i>} = 0.5. In <b>A</b>: Divisive rescaling of the input/output relation with <i>g</i>_<i>A</i> = 20. In <b>B</b>: Subtractive shifting of the input/output relation with <i>g</i>_<i>A</i> = 40. <b>C</b>: Data from <b>A</b> and <b>B</b> are replotted with output firing rates in the absence of inhibition on the ordinate and output firing rates in the presence of inhibition on the abscissa. Threshold-linear functions are fit to simulation data (black lines). Rightward shift of threshold-linear function for <i>g</i>_<i>A</i> = 40 is characteristic of subtractive inhibition.</p

Approximation of <i>b</i> (a slow variable) by its average value.

<p><b>A</b>: Plots of <i>b</i>_<i>av</i> vs. <i>r</i>_<i>E</i> for different values of <i>g</i>_<i>A</i>. <b>B</b>: Plots of , and <i>b</i>_<i>av</i>(<i>r</i>_<i>E</i>, <i>g</i>_<i>A</i>) with <i>g</i>_<i>A</i> = 40. In both panels, <i>g</i>_{<i>Syn</i>,<i>E</i>} = 3, <i>g</i>_{<i>Syn</i>,<i>I</i>} = 5 and <i>r</i>_<i>I</i> = 50 Hz.</p

Output firing rates approximated as dead-time modified Poisson process with firing threshold.

<p><b>Top row</b>: Firing rate as a function of input rate obtained from simulations (circles) and theoretical approximation (lines); for <i>g</i>_<i>A</i> = 15 (A1), <i>g</i>_<i>A</i> = 25 (B1), and <i>g</i>_<i>A</i> = 35 (C1). Simulations and theory show transition from divisive to subtractive inhibition as <i>g</i>_<i>A</i> increases. <b>Bottom row</b>: Theoretical approximation of firing threshold <i>θ</i> (lines), and the largest observed values of <i>s</i>_<i>I</i> for which excitatory inputs elicited spikes in simulations (circles), plotted as functions of input rate; for <i>g</i>_<i>A</i> = 15 (A2), <i>g</i>_<i>A</i> = 25 (B2), and <i>g</i>_<i>A</i> = 35 (C2).</p