The voltage and spiking responses of subthreshold resonant neurons to structured and fluctuating inputs: persistence and loss of resonance and variability

Abstract

We systematically investigate the response of neurons to oscillatory currents and synaptic-like inputs and we extend our investigation to non-structured synaptic-like spiking inputs with more realistic distributions of presynaptic spike times. We use two types of chirp-like inputs consisting of (i) a sequence of cycles with discretely increasing frequencies over time, and (ii) a sequence having the same cycles arranged in an arbitrary order. We develop and use a number of frequency-dependent voltage response metrics to capture the different aspects of the voltage response, including the standard impedance (Z) and the peak-to-trough amplitude envelope ($V_{\text {ENV}}$) profiles. We show that Z-resonant cells (cells that exhibit subthreshold resonance in response to sinusoidal inputs) also show $V_{\text {ENV}}$-resonance in response to sinusoidal inputs, but generally do not (or do it very mildly) in response to square-wave and synaptic-like inputs. In the latter cases the resonant response using Z is not predictive of the preferred frequencies at which the neurons spike when the input amplitude is increased above subthreshold levels. We also show that responses to conductance-based synaptic-like inputs are attenuated as compared to the response to current-based synaptic-like inputs, thus providing an explanation to previous experimental results. These response patterns were strongly dependent on the intrinsic properties of the participating neurons, in particular whether the unperturbed Z-resonant cells had a stable node or a focus. In addition, we show that variability emerges in response to chirp-like inputs with arbitrarily ordered patterns where all signals (trials) in a given protocol have the same frequency content and the only source of uncertainty is the subset of all possible permutations of cycles chosen for a given protocol. This variability is the result of the multiple different ways in which the autonomous transient dynamics is activated across cycles in each signal (different cycle orderings) and across trials. We extend our results to include high-rate Poisson distributed current- and conductance-based synaptic inputs and compare them with similar results using additive Gaussian white noise. We show that the responses to both Poisson-distributed synaptic inputs are attenuated with respect to the responses to Gaussian white noise. For cells that exhibit oscillatory responses to Gaussian white noise (band-pass filters), the response to conductance-based synaptic inputs are low-pass filters, while the response to current-based synaptic inputs may remain band-pass filters, consistent with experimental findings. Our results shed light on the mechanisms of communication of oscillatory activity among neurons in a network via subthreshold oscillations and resonance and the generation of network resonance.

Appendix: Intrinsic and resonant oscillatory properties of 2D linear systems

Consider

$$\begin{aligned} \left\{ \begin{array}{ll} x' = a\, x + b\, y + A_{in}\, e^{i\, \omega \, t},\\ y' = c\, x+ d\, y \end{array} \right. \end{aligned}$$

(15)

where a, b, c and d are constants, $\omega = 2 \pi f / 1000 > 0$ is the input frequency and $A_{in} \ge 0$ is the input amplitude. The prime sign represents the derivative with respect to t. The units of t are ms and the units of f are Hz.

1.1 Intrinsic oscillations

The characteristic polynomial for the corresponding homogeneous system ($A_{in} = 0$) is given by

$$\begin{aligned} r^2 - (a + d)\, r+ ( a\, d - b\, c ) = 0. \end{aligned}$$

(16)

The eigenvalues are given by

$$\begin{aligned} r_{1,2} = \frac{a+d \pm \sqrt{(a-d)^2+4 b c}}{2}, \end{aligned}$$

(17)

and the natural (intrinsic) frequency of the (damped) oscillations (in Hz if t has units of ms) is given by

$$\begin{aligned} f_{nat} = \frac{\sqrt{-(a-d)^2-4 b c}}{4 \pi }\, 1000 \end{aligned}$$

(18)

assuming $(a-d)^2+4 b c < 0$.

1.2 Resonance and the impedance amplitude profile

The impedance amplitude profile $Z(\omega )$ for system (15)–(16) is the magnitude

$$\begin{aligned} Z(\omega ) = \sqrt{\frac{d^2 + \omega ^2}{(a\, d - b\, c - \omega ^2)^2 + (a+d)^2\, \omega ^2}} \end{aligned}$$

(19)

of the complex valued coefficient of the particular solution to the system

$$\begin{aligned} \mathbf{Z}(\omega ) = \frac{(-d+i\, \omega )}{(-a\, + i\, \omega )\, (-d+i\, \omega ) - b\, c}. \end{aligned}$$

(20)

For 1D system, these quantities are given, respectively, by

$$\begin{aligned} Z(\omega ) = \frac{1}{\sqrt{a^2+\omega ^2}} \end{aligned}$$

(21)

and

$$\begin{aligned} \mathbf{Z}(\omega ) = \frac{1}{(-a\, + i\, \omega )}. \end{aligned}$$

(22)

The resonance frequency $f_{\text {res}}$ (in Hz if t has units of ms) is the frequency at which Z reaches its maximum

$$\begin{aligned} f_{\text {res}} = \frac{\sqrt{ -d^2 + \sqrt{b^2\, c^2 - 2\, a\, b\, c\, d - 2\, d^2\, b\, c}}}{2 \pi }\, 1000. \end{aligned}$$

(23)