Aperiodic stochastic resonance in neural information processing with Gaussian colored noise

Abstract

The aim of this paper is to explore the phenomenon of aperiodic stochastic resonance in neural systems with colored noise. For nonlinear dynamical systems driven by Gaussian colored noise, we prove that the stochastic sample trajectory can converge to the corresponding deterministic trajectory as noise intensity tends to zero in mean square, under global and local Lipschitz conditions, respectively. Then, following forbidden interval theorem we predict the phenomenon of aperiodic stochastic resonance in bistable and excitable neural systems. Two neuron models are further used to verify the theoretical prediction. Moreover, we disclose the phenomenon of aperiodic stochastic resonance induced by correlation time and this finding suggests that adjusting noise correlation might be a biologically more plausible mechanism in neural signal processing.

Appendix

Proof of Lemma 1

Proof

Fix $T \ge 0$ arbitrarily. The Ito formula (Øksendal 2005; Mao 2007) shows that

$$\begin{aligned} & \left| {u_{j} (t)} \right|^{2k} = \left| {u_{j} (0)} \right|^{2k} + \int_{0}^{t} {\left( { - \frac{2k}{\tau }\left| {u_{j} (s)} \right|^{2k} + k(2k - 1)\sigma^{2} \left| {u_{j} (s)} \right|^{2(k - 1)} } \right)ds} \\ & \quad + 2k\sigma \int_{0}^{t} {(u_{j} (s))^{2k - 1} } dW_{j} (s) \\ \end{aligned}$$

for $0 \le t \le T$. By the moment property (3) of the stationary OU process, we get

$$E\left[ {\;\mathop {\sup }\limits_{0 \le t \le T} \left| {u_{j} (t)} \right|^{2k} } \right] \le \sigma^{2k} (2k - 1)!!(0.5\tau )^{k - 1} (0.5\tau + kT) + 2k\sigma E\left[ {\;\mathop {\sup }\limits_{0 \le t \le T} \int_{0}^{t} {(u_{j} (s))^{2k - 1} dW_{j} (s)} } \right]$$

By the Burkholder–Davis–Gundy inequality (Prato and Zabczyk 1992),

$$E\left[ {\;\mathop {\sup }\limits_{0 \le t \le T} \left| {u_{j} (t)} \right|^{2k} } \right] \le \sigma^{2k} (2k - 1)!!(0.5\tau )^{k - 1} (0.5\tau + kT) + 2\sqrt 3 k\sigma E\left[ {\left( {\int_{0}^{T} {\left| {u_{j} (s)} \right|^{4k - 2} ds} } \right)^{{\frac{1}{2}}} } \right].$$

Using the Hölder inequality we then derive

$$\begin{aligned} E\left[ {\;\mathop { \sup }\limits_{0 \le t \le T} \left| {u_{j} (t )} \right|^{2k} } \right] & \le \sigma^{2k} (2k - 1 )!! (0.5\tau )^{k - 1} (0.5\tau + kT )\\ & \quad + 2\sqrt 3 k\sigma \left( {\int_{0}^{T} {E\left[ {\;\left| {u_{j} (s )} \right|^{4k - 2} } \right]ds} } \right)^{{\frac{1}{2}}} \\ & \le \sigma^{2k} (2k - 1 )!! (0.5\tau )^{k - 1} (0.5\tau + kT )\\ & \quad + 2\sqrt 3 k\sigma \left( {T (4k - 3 )!! (0.5\tau \sigma^{2} )^{2k - 1} } \right)^{{\frac{1}{2}}} \\ & \le \sigma^{2k} \left( { (0.5\tau )^{k - 1} (0.5\tau + kT )+ 2k\sqrt {3T (4k - 3 )!! (0.5\tau )^{2k - 1} } } \right). \\ \end{aligned}$$

□

Proof of Lemma 2

Proof

It is well known that almost all sample paths of the Ornstein–Ulenbeck process are continuous. It is therefore easy to see from the classical theory of ordinary differential equations that for any initial value $X_{0} \in R^{d}$, Eq. (1) has a unique global solution $X_{t}$ on $t \ge 0$. Fix $T \ge 0$ arbitrarily. According to Lemma 1,

$$E\left[ {\;\mathop {\sup }\limits_{0 \le t \le T} \left| {u_{j} (t)} \right|^{2k} } \right] \le \sigma^{2k} \xi_{k}$$

(30)

with $\xi_{k}$ given by Eq. (7).

Define the stop** times $\tau_{h} = \inf \{ t \ge 0:\left| {X_{t} } \right| \ge h\}$ for all integers $h > \left| {X_{0} } \right|$, where throughout this paper we set $\inf \varPhi = \infty$. Here $\varPhi$ stands for the empty set. Clearly, $\tau_{h} \to \infty$ almost surely as $h \to \infty$. For $t \in \left[ {0,T} \right]$, it follows from Eq. (1a) that

$$\begin{aligned} \left| {X_{{t \wedge \tau_{h} }}^{i} } \right|^{p} & \le (m + 2)^{p - 1} \left( {\left| {X_{0}^{i} } \right|^{p} + \left| {\int_{0}^{{t \wedge \tau_{h} }} {f^{i} (X_{s} ,s)ds} } \right|^{p} + \sum\limits_{j = 1}^{m} {\left| {\int_{0}^{{t \wedge \tau_{h} }} {g_{j}^{i} (X_{s} ,s)u_{j} (s)ds} } \right|^{p} } } \right) \\ & \le (m + 2)^{p - 1} \left( {\left| {X_{0}^{i} } \right|^{p} + t^{p - 1} \int_{0}^{{t \wedge \tau_{h} }} {\left| {f^{i} (X_{s} ,s)} \right|^{p} ds} + t^{p - 1} \sum\limits_{j = 1}^{m} {\int_{0}^{{t \wedge \tau_{h} }} {\left| {g_{j}^{i} (X_{s} ,s)u_{j} (s)} \right|^{p} ds} } } \right) \\ & \le (m + 2)^{p - 1} \left( {\left| {X_{0}^{i} } \right|^{p} + t^{p - 1} \int_{0}^{t} {\left| {f^{i} \left( {X_{{s \wedge \tau_{h} }} ,s \wedge \tau_{h} } \right)} \right|^{p} ds} + t^{p - 1} \sum\limits_{j = 1}^{m} {\int_{0}^{t} {\left| {g_{j}^{i} \left( {X_{{s \wedge \tau_{h} }} ,s \wedge \tau_{h} } \right)u_{j} (s \wedge \tau_{h} )} \right|^{p} ds} } } \right) \\ & \le (m + 2)^{p - 1} \left( {\left| {X_{0}^{i} } \right|^{p} + t^{p - 1} K^{p} \int_{0}^{t} {\left( {1 + X_{{s \wedge \tau_{h} }} } \right)^{p} ds} + t^{p - 1} K^{p} \sum\limits_{j = 1}^{m} {\int_{0}^{t} {\left( {1 + \left| {X_{{s \wedge \tau_{h} }} } \right|^{\gamma } } \right)^{p} \left| {u_{j} (s \wedge \tau_{h} )} \right|^{p} ds} } } \right) \\ & \le (m + 2)^{p - 1} \left( {\left| {X_{0}^{i} } \right|^{p} + t^{p - 1} 2^{p - 1} K^{p} \int_{0}^{t} {\left( {1 + \left| {X_{{s \wedge \tau_{h} }} } \right|^{p} } \right)ds} + t^{p - 1} 2^{p - 1} K^{p} \sum\limits_{j = 1}^{m} {\int_{0}^{t} {\left( {1 + \left| {X_{{s \wedge \tau_{h} }} } \right|^{p\gamma } } \right)\left| {u_{j} (s \wedge \tau_{h} )} \right|^{p} ds} } } \right) \\ \end{aligned}$$

Here, the first inequality is due to $(a_{1} + \cdots + a_{m} )^{p} \le m^{p - 1} (\left| {a_{1} } \right|^{p} + \cdots + \left| {a_{m} } \right|^{p} )$, the second inequality is owing to the Hölder inequality; the growth conditions are adopted for the last second equality; and the inequality $(\left| a \right| + \left| b \right| )^{p} \le 2^{p - 1} (\left| a \right|^{p} + \left| b \right|^{p} )$ is used in the last inequality. As the right-hand-side terms are increasing in $t$, we see easily that

$$\begin{aligned} & E\left[ {\;\mathop {\sup }\limits_{0 \le s \le t} \left| {X_{{s \wedge \tau_{h} }}^{i} } \right|^{p} } \right] \le (m + 2)^{p - 1} \left( {\left| {X_{0}^{i} } \right|^{p} + T^{p - 1} 2^{p - 1} K^{p} \times}\right.\\&\quad \left.{\int_{0}^{t} {\left( {1 + E\left[ {\;\left| {X_{{s \wedge \tau_{h} }} } \right|^{p} } \right]} \right)ds} } \right) + (m + 2)^{p - 1} T^{p - 1} 2^{p - 1} K^{p}\times\\ & \quad \sum\limits_{j = 1}^{m} {\int_{0}^{t} {E\left[ {\;\left( {1 + \left| {X_{{s \wedge \tau_{h} }} } \right|^{p\gamma } } \right)\left| {u_{j} (s \wedge \tau_{h} )} \right|^{p} } \right]ds} } \\ \end{aligned}$$

and then by $\left| {X_{ 0}^{i} } \right|^{p} = \left( {\left| {X_{ 0}^{i} } \right|^{ 2} } \right)^{{\frac{p}{2}}} \le \left( {\sum\limits_{i = 1}^{d} {\left| {X_{ 0}^{i} } \right|^{ 2} } } \right)^{{\frac{p}{2}}} = \left| {X_{0} } \right|^{p} ,$

$$\begin{aligned} & E\left[ {\;\mathop {\sup }\limits_{0 \le s \le t} \left| {X_{{s \wedge \tau_{h} }}^{i} } \right|^{p} } \right] \le (m + 2)^{p - 1} \left( {\left| {X_{0} } \right|^{p} + T^{p - 1} 2^{p - 1} K^{p} \times}\right.\\&\quad\left.{\int_{0}^{t} {\left( {1 + E\left[ {\;\mathop {\sup }\limits_{0 \le r \le s} \left| {X_{{r \wedge \tau_{h} }} } \right|^{p} } \right]} \right)ds} } \right) \\ & \quad + (m + 2)^{p - 1} T^{p - 1} 2^{p - 1} K^{p} \sum\limits_{j = 1}^{m} \\&\quad{\int_{0}^{t} {E\left[ {\;\left( {1 + \mathop {\sup }\limits_{0 \le r \le s} \left| {X_{{r \wedge \tau_{h} }} } \right|^{p\gamma } } \right)\mathop {\sup }\limits_{0 \le r \le s} \left| {u_{j} (r \wedge \tau_{h} )} \right|^{p} } \right]ds} } \\ \end{aligned}$$

By the well-known Young inequality $xy \le \frac{{x^{p} }}{p} + \frac{{y^{q} }}{q}$ for $x,y \ge 0$ and $p,q > 0$ with $\frac{1}{p} + \frac{1}{q} = 1$,

$$\begin{aligned} & E\left[ {\;\mathop {\sup }\limits_{0 \le r \le s} \left| {X_{{r \wedge \tau_{h} }} } \right|^{p\gamma } \mathop {\sup }\limits_{0 \le r \le s} \left| {u_{j} (r \wedge \tau_{h} )} \right|^{p} } \right] \le \gamma E\left[ {\mathop {\sup }\limits_{0 \le r \le s} \left| {X_{{r \wedge \tau_{h} }} } \right|^{p} } \right] \\ & \quad \quad + (1 - \gamma )E\left[ {\;\mathop {\sup }\limits_{0 \le r \le s} \left| {u_{j} (r \wedge \tau_{h} )} \right|^{{\frac{p}{1 - \gamma }}} } \right] \\ & \quad \le E\left[ {\mathop {\sup }\limits_{0 \le r \le s} \left| {X_{{r \wedge \tau_{h} }} } \right|^{p} } \right] + E\left[ {\;\mathop {\sup }\limits_{0 \le r \le s} \left| {u_{j} (r \wedge \tau_{h} )} \right|^{{\frac{p}{1 - \gamma }}} } \right] \\ \end{aligned}$$

while recalling that $\bar{k} \ge \frac{p}{2(1 - \gamma )}$ in Eq. (13), then by the Hölder inequality,

$$\begin{aligned} & E\left[ {\;\mathop {\sup }\limits_{0 \le r \le s} \left| {u_{j} (r \wedge \tau_{h} )} \right|^{{\frac{p}{1 - \gamma }}} } \right] \le E\left[ {\;\mathop {\sup }\limits_{0 \le r \le s} \left| {u_{j} (r \wedge \tau_{h} )} \right|^{{ 2\bar{k}}} } \right]^{{\frac{p}{{2(1 - \gamma )\bar{k}}}}} \\ & \quad \le E\left[ {\;\mathop {\sup }\limits_{0 \le r \le T} \left| {u_{j} (r \wedge \tau_{h} )} \right|^{{ 2\bar{k}}} } \right]^{{\frac{p}{{2(1 - \gamma )\bar{k}}}}} \\ \end{aligned}$$

Hence, by Eq. (30),

$$\begin{aligned} & E\left[ {\;\mathop {\sup }\limits_{0 \le s \le t} \left| {X_{{s \wedge \tau_{h} }}^{i} } \right|^{p} } \right] \le (m + 2)^{p - 1} \\ & \quad (\left| {X_{0} } \right|^{p} + T^{p - 1} 2^{p - 1} K^{p} \left( {1 + m\sigma^{p} \xi_{p}^{{\frac{1}{2}}} + m\sigma^{{\frac{p}{1 - \gamma }}} \xi_{{\bar{k}}}^{{\frac{p}{{2(1 - \gamma )\bar{k}}}}} } \right) \\ & \quad + (m + 2)^{p - 1} T^{p - 1} 2^{p - 1} K^{p} (m + 1)\int_{0}^{t} {E\left[ {\;\mathop {\sup }\limits_{0 \le r \le s} \left| {X_{{r \wedge \tau_{h} }} } \right|^{p} } \right]ds} \\ \end{aligned}$$

Considering

$$\begin{aligned} \mathop {\sup }\limits_{0 \le s \le t} \left| {X_{{s \wedge \tau_{h} }} } \right|^{p} & = \mathop {\sup }\limits_{0 \le s \le t} \left( {\sum\limits_{i = 1}^{d} {\left| {X_{{s \wedge \tau_{h} }}^{i} } \right|^{2} } } \right)^{{\frac{p}{2}}} \le \left( {d\mathop {\hbox{max} }\limits_{1 \le i \le d} \mathop {\sup }\limits_{0 \le s \le t} \left| {X_{{s \wedge \tau_{h} }}^{i} } \right|^{2} } \right)^{{\frac{p}{2}}} \\ & = d^{{^{{\frac{p}{2}}} }} \mathop {\hbox{max} }\limits_{1 \le i \le d} \mathop {\sup }\limits_{0 \le s \le t} \left| {X_{{s \wedge \tau_{h} }}^{i} } \right|^{p} , \\ \end{aligned}$$

then

$$E\left[ {\mathop {\sup }\limits_{0 \le s \le t} \left| {X_{{s \wedge \tau_{h} }} } \right|^{p} } \right] \le d^{{^{{\frac{p}{2}}} }} E\left[ {\mathop {\hbox{max} }\limits_{1 \le i \le d} \mathop {\sup }\limits_{0 \le s \le t} \left| {X_{{s \wedge \tau_{h} }}^{i} } \right|^{p} } \right] \le d^{{^{{\frac{p}{2} + 1}} }} \mathop {\hbox{max} }\limits_{1 \le i \le d} E\left[ {\mathop {\sup }\limits_{0 \le s \le t} \left| {X_{{s \wedge \tau_{h} }}^{i} } \right|^{p} } \right].$$

Here, the distribution property for the maximum of multiple mutually independent random variables is adopted. Then for any $0 \le t \le T$,

$$E\left[ {\mathop {\sup }\limits_{0 \le s \le t} \left| {X_{{s \wedge \tau_{h} }} } \right|^{p} } \right] \le a_{p} + b_{p} \int_{0}^{t} {E\left[ {\mathop {\sup }\limits_{0 \le r \le s} \left| {X_{{r \wedge \tau_{h} }} } \right|^{p} } \right]} ds$$

(31)

with $a_{p}$ and $b_{p}$ given in Eqs. (11) and (12). And then, the application of the Gronwall inequality to Eq. (31) yields

$$E\left[ {\mathop {\sup }\limits_{0 \le s \le T} \left| {X_{{s \wedge \tau_{h} }} } \right|^{p} } \right] \le a_{p} \exp (b_{p} T) < \infty$$

Letting $h \to \infty$ implies the required assertion (10). □

Proof of Lemma 3

Proof

Note the inequality (15) can be proven with technique somehow parallel to that of Lemma 2. It is well known that under given conditions Eq. (3) has a unique global solution $\hat{X}_{t}$ on $t \ge 0$. Define a sequence $v_{h} = \inf \{ t \ge 0:\left| {\hat{X}_{t} } \right| \ge h\}$ for all integers $h \ge \left| {X_{0} } \right|$, with $\inf \varPhi = \infty$ for an empty set $\varPhi$. Clearly, $v_{h} \to \infty$ almost surely as $h \to \infty$. For $t \in \left[ {0,T} \right]$, it can be deduced from (3) that for $0 < t < T$,

$$\mathop {\sup }\limits_{0 \le s \le t} \left| {\hat{X}_{{s \wedge v_{h} }} } \right|^{p} \le d^{{\frac{p}{2}}} (2^{p - 1} \left| {X_{0} } \right|^{p} + T^{p} 2^{2(p - 1)} K^{p} ) + T^{p - 1} 2^{2(p - 1)} d^{{\frac{p}{2}}} K^{p} \int_{0}^{t} {\mathop {\sup }\limits_{0 \le r \le s} \left| {\hat{X}_{{r \wedge v_{h} }} } \right|^{p} ds}$$

Then, the Gronwall inequality implies

$$\mathop {\sup }\limits_{0 \le s \le t} \left| {\hat{X}_{{s \wedge v_{h} }} } \right|^{p} \le d^{{\frac{p}{2}}} (2^{p - 1} \left| {X_{0} } \right|^{p} + T^{p} 2^{2(p - 1)} K^{p} )\exp \left( {2^{2(p - 1)} d^{{\frac{p}{2}}} T^{p} } \right)$$

Letting $h \to \infty$ implies the assertion (15) immediately. □

Proof of Lemma 4

Proof

Recalling the duplicate property of the conditional probability distribution

$$E\left[ {E\left[ {\mathop {\sup }\limits_{0 \le t \le T} \left| {X_{t} - \hat{X}_{t} } \right|^{2} \left| {S = s_{1} } \right.} \right]} \right] = E\left[ {\mathop {\sup }\limits_{0 \le t \le T} \left| {X_{t} - \hat{X}_{t} } \right|^{2} } \right],$$

we obtain

$$\begin{aligned} &E\left[ {\mathop {\sup }\limits_{0 \le t \le T} \left| {X_{t} - \hat{X}_{t} } \right|^{2} } \right]\\& = P\left\{ {S = s_{1} } \right\}E\left[ {\mathop {\sup }\limits_{0 \le t \le T} \left| {X_{t} - \hat{X}_{t} } \right|^{2} \left| {S = s_{1} } \right.} \right]\\& \quad+ P\left\{ {S = s_{2} } \right\}E\left[ {\mathop {\sup }\limits_{0 \le t \le T} \left| {X_{t} - \hat{X}_{t} } \right|^{2} \left| {S = s_{2} } \right.} \right],\end{aligned}$$

from which it can be deduced that

$$E\left[ {\mathop {\sup }\limits_{0 \le t \le T} \left| {X_{t} - \hat{X}_{t} } \right|^{2} \left| {S = s_{i} } \right.} \right] \le \frac{1}{{P\left\{ {S = s_{i} } \right\}}}E\left[ {\mathop {\sup }\limits_{0 \le t \le T} \left| {X_{t} - \hat{X}_{t} } \right|^{2} } \right],$$

and thus by Theorem 2, Eq. (25) is found true. Then, application of Markov’s inequality immediately gives Eq. (26). □