Oscillations and variability in neuronal systems: interplay of autonomous transient dynamics and fast deterministic fluctuations

Abstract

Neuronal systems are subject to rapid fluctuations both intrinsically and externally. These fluctuations can be disruptive or constructive. We investigate the dynamic mechanisms underlying the interactions between rapidly fluctuating signals and the intrinsic properties of the target cells to produce variable and/or coherent responses. We use linearized and non-linear conductance-based models and piecewise constant (PWC) inputs with short duration pieces. The amplitude distributions of the constant pieces consist of arbitrary permutations of a baseline PWC function. In each trial within a given protocol we use one of these permutations and each protocol consists of a subset of all possible permutations, which is the only source of uncertainty in the protocol. We show that sustained oscillatory behavior can be generated in response to various forms of PWC inputs independently of whether the stable equilibria of the corresponding unperturbed systems are foci or nodes. The oscillatory voltage responses are amplified by the model nonlinearities and attenuated for conductance-based PWC inputs as compared to current-based PWC inputs, consistent with previous theoretical and experimental work. In addition, the voltage responses to PWC inputs exhibited variability across trials, which is reminiscent of the variability generated by stochastic noise (e.g., Gaussian white noise). Our analysis demonstrates that both oscillations and variability are the result of the interaction between the PWC input and the target cell’s autonomous transient dynamics with little to no contribution from the dynamics in vicinities of the steady-state, and do not require input stochasticity.

Appendices

Intrinsic and resonant oscillatory properties of 2D linear systems

Consider

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

(10)

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

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

(11)

The eigenvalues are given by

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

(12)

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

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

(13)

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

1.2 Resonance and the impedance amplitude profile

The impedance amplitude profile $Z(\omega )$ for system (10)-(11) is the magnitude

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

(14)

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

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

(15)

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

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

(16)

and

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

(17)

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

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

(18)

1.3 Response to constant inputs

The equilibrium solution to system (10) for a constant input $A_{in}$ (i.e., $\omega = 0$) is given by

$$\begin{aligned} \bar{x} = -\frac{A_{in}\, d}{a d - b c} \quad \text{and} \quad \bar{y} = \frac{A_{in}\, c}{a d - b c}. \end{aligned}$$

(19)

The eigenvectors are given by

$$\begin{aligned} z_{1,2} = [b\ \ (r_{1,2}-a)]^T. \end{aligned}$$

(20)

The solution satisfying the initial conditions $[x(0)\ \ y(0)]^T = [x_0\ \ y_0]^T$ is given by

$$\begin{aligned} \left[ \begin{array}{lll} x \\ y \end{array} \right] = c_1\, \left[ \begin{array}{lll} \ \ \ \ \ b \\ r_1 -a \end{array} \right] \, e^{r_1 t} + c_2\, \left[ \begin{array}{lll} \ \ \ \ \ b \\ r_2 -a \end{array} \right] \, e^{r_2 t} + \left[ \begin{array}{lll} \bar{x} \\ \bar{y} \end{array} \right] \end{aligned}$$

(21)

where

$$\begin{aligned} &c_1 = \frac{b\, (y_0 - \bar{y}) - (x_0 - \bar{x})\, (r_2 - a)}{b\, (r_1 - r_2)} \quad \text{ and } \\& c_2 = \frac{-b\, (y_0 - \bar{y}) + (x_0 - \bar{x})\, (r_1 - a)}{b\, (r_1 - r_2)} \end{aligned}$$

(22)

For 1D systems ($b = 0$),

$$\begin{aligned} \bar{x} = -\frac{A_{in}}{a} \end{aligned}$$

(23)

and

$$\begin{aligned} x = \left( x_0 - \frac{A_{in}}{a} \right) e^{a t} -\frac{A_{in}}{a}, \end{aligned}$$

(24)

where $x(0) = x_0$

Ornstein-uhlenbeck (OU) process

1.1 One-dimensional OU process

The 1D OU process Uhlenbeck and Ornstein (1930) is described by the following linear stochastic differential equation

$$\begin{aligned} X' = -a X + I + \sigma \eta (t) \end{aligned}$$

(25)

where $a>0$, I and $\sigma$ are constants and $\eta (t)$ is zero-mean and $\delta$-correlated Gaussian white noise. The parameter a is the inverse of the time constant and measure the strength by which the system reacts to perturbations. The parameter $\sigma$ measures the intensity of the noise. The quotient $I/\alpha$ is the asymptotic mean.

Using standard methods Risken (1989); Gardiner (1985) one can compute the solution satisfying $X(0) = x_0$, which is the sum of a deterministic function with the form (24) and an integral of a deterministic function with respect to a Wiener process. The solution is normally distributed with mean and variance given, respectively by

$$\begin{aligned} E[X(t)] = \left( X_0 - \frac{I}{a} \right) e^{-a t} +\frac{I}{a} \end{aligned}$$

(26)

and

$$\begin{aligned} Var[X(t)] = \frac{\sigma ^2}{2 a}\, (1 - e^{-2 a t}). \end{aligned}$$

(27)

1.2 Higher-dimensional OU process

The multivariate OU process Uhlenbeck and Ornstein (1930) is described by the following linear stochastic differential equation

$$\begin{aligned} X' = A X + B + \Sigma H(t) \end{aligned}$$

(28)

where X is an n-dimensional vector, A is an $n \times n$ matrix, B is an n-dimensional vector, $\Sigma$ is an $n \times m$ matrix and H is a vector of independent zero-mean and $\delta$-correlated Gaussian white noise components. Using standard methods Risken (1989); Gardiner (1985) one can compute the solution satisfying $X(0) = x_0$. The solution is normally distributed. The mean is given by

$$\begin{aligned} E[X(t)] = [\, e^{A t} - B\, ]\, A^{-1} B + e^{A t} X(0), \end{aligned}$$

(29)

and the covariance matrix is given by

$$\begin{aligned} Cov[X(t)] = \int _0^t e^{A s} \Sigma \Sigma ^T e^{A^T s}. \end{aligned}$$

(30)

Under certain conditions, the covariance matrix corresponding to the stationary solutions reads

$$\begin{aligned} Cov[X(t)] = -\frac{\Sigma \Sigma ^T}{2\, Tr(A)} - \frac{[A - Tr(A) I]\, \Sigma \Sigma ^T\, [A - Tr(A) I]^T}{2\, Tr(A)\, det(A)}. \end{aligned}$$

(31)