Neurons as ideal change-point detectors

Abstract

Every computational unit in the brain monitors incoming signals, instant by instant, for meaningful changes in the face of stochastic fluctuation. Recent studies have suggested that even a single neuron can detect changes in noisy signals. In this paper, we demonstrate that a single leaky integrate-and-fire neuron can achieve change-point detection close to that of theoretical optimal, for uniform-rate process, functions even better than a Bayes-optimal algorithm when the underlying rate deviates from a presumed uniform rate process. Given a reasonable number of synaptic connections (order 10⁴) and the rate of the input spike train, the values of the membrane time constant and the threshold found for optimizing change-point detection are close to those seen in biological neurons. These findings imply that biological neurons could act as sophisticated change-point detectors.

Appendices

Appendix A: Monte Carlo procedure for computing a risk function

In accordance with the definition given in Eqs. (1) and (2), the risk function R is numerically computed as follows:

• Iterate the following (i)–(iii) N times:

  • (i) Generate a change-point θ _j from the distribution $p(\theta ) = {\bar{\theta }^{{ - 1}}}\exp (\theta /\bar{\theta })$.

  • (ii) Generate a Poisson spike train X _t whose rate changes from λ ⁽⁰⁾ to λ ⁽¹⁾ at time θ _j , and simulate a detection time T _j by simulating Eq. (4).

  • (iii) Compute a single trial risk ${U_j} = U({T_j},{\theta_j})$.

• Let $R = \sum\nolimits_{{j = 1}}^N {{U_j}/N}$.

The iteration number, N, should be large enough for an accurate computation. In this study we set N at 10⁵.

Appendix B: Approximation method for computing a risk function

To develop a fast method for computing the risk function, we approximate Eq. (4) to a diffusion process with the same first and second infinitesimal moments as in Eq. (4), i.e.,

$$ d{V_t} = - \frac{1}{\tau }{V_t}dt + d{X_t} \approx \mu ({V_t})dt + \sigma ({V_t})d{W_t}, $$

(7)

where μ(v), σ²(v), and W _t denote the first and the second infinitesimal moments and the standard Wiener process, respectively. This diffusion approximation holds true in the limit of small amplitude of individual steps in the voltage and high frequency of spike arrival (Walsh 1981; Tuckwell 1988). From the relations that $E[d{X_t}] = {\lambda^{{(i)}}}dt,\,E[d{X_t}d{X_{{t\prime}}}] = {\lambda^{{(i)}}}dt{\lambda^{{(i)}}}dt\prime + \delta (t - t\prime){\lambda^{{(i)}}}dtdt\prime$, (i = 0 for $t,t\prime < \theta$, i = 1 for $t,t\prime \geqslant \theta$), and ${V_t} = v\exp ( - t/\tau ) + \int_0^t {\exp ( - s/\tau )d{X_{{t - s}}}}$ on the condition ${V_{{t = 0}}} = v$, we obtain the infinitesimal moments as follows:

$$ \begin{gathered} \mu (v) = \mathop{{\lim }}\limits_{{t \to 0}} \frac{1}{t}E[{V_t} - v] = - \frac{1}{\tau }(v - \tau {\lambda^{{(i)}}})\,, \\ {\sigma^2}(v) = \mathop{{\lim }}\limits_{{t \to 0}} \frac{1}{t}E\left[ {{{({V_t} - E({V_t}))}^2}} \right] = {\lambda^{{(i)}}}. \\ \end{gathered} $$

(8)

The stochastic differential equation in Eq. (7) is thus

$$ d{V_t} = - \frac{1}{\tau }({V_t} - \tau {\lambda_t})dt + \sqrt {{{\lambda_t}}} d{W_t}, $$

(9)

where λ ⁽ⁱ⁾ is replaced by λ _t for simplicity. Equation (9) is an Ornstein-Uhlenbeck process (Gardiner 2009), which is a well-studied stochastic process. Because the time of detection T is the time when V _t rises above threshold S for the first time, the distribution of T is equivalent to the first-passage-time distribution of Eq. (9). We denote it by p _0→S (t), where the subscript a → b represents the initial state and the threshold of V _t as a and b, respectively. According to the theory of stochastic differential equations (Gardiner 2009), p _0→S (t) can be expressed using the probability density of V _t , denoted by P(V, t), as follows:

$$ {p_{{0 \to S}}}(t) = - \frac{\partial }{{\partial t}}\int_{{ - \infty }}^S {P(V,t)dV} $$

(10)

where P(V, t) satisfies the following partial differential (Fokker-Plank) equation for Eq. (9),

$$ \frac{{\partial P(V,t)}}{{\partial t}} = \frac{1}{2}{\lambda_t}\frac{{{\partial^2}P}}{{\partial {V^2}}} + \frac{1}{\tau }\frac{{\partial \left[ {(V - \tau {\lambda_t})P} \right]}}{{\partial V}} $$

(11)

under the boundary conditions

$$ \begin{gathered} P(V,0) = \delta (V)\,, \\ P(S,t) = 0\,, \\ \frac{{\partial P( - \infty, t)}}{{\partial V}} = 0\,. \\ \end{gathered} $$

(12)

Now, we express the risk function in Eq. (2) using p _0→S (t) and the distribution of change-point p (θ) as,

$$ R = \int {d\theta p(\theta )\left[ {\int_0^{\theta } {{p_{{0 \to S}}}(T)dT} + c\int_{\theta }^{\infty } {(T - \theta ){p_{{0 \to S}}}(T)dT} } \right]} \,. $$

(13)

From Eq. (10), the first term of Eq. (13) can be written as follows:

$$ \begin{gathered} \int {d\theta p(\theta )} \int_0^{\theta } {dt\left( { - \frac{\partial }{{\partial t}}\int_{{ - \infty }}^S {P(V,t)dV} } \right)} \hfill \\ = 1 - \int_{{ - \infty }}^S {f(V)dV}, \hfill \\ \end{gathered} $$

(14)

where

$$ f(V) = \int_0^{\infty } {p(\theta )P(V,\theta )d\theta } . $$

(15)

From Eqs. (11) and (12), f (V) satisfies the following equation,

$$ \begin{gathered} \frac{{\bar{\theta }{\lambda^{{(0)}}}}}{2}\frac{{{d^2}f(V)}}{{d{V^2}}} + \frac{{\bar{\theta }}}{\tau }(V - \tau {\lambda^{{(0)}}})\frac{{df}}{{dV}} + \left( {\frac{{\bar{\theta }}}{\tau } - 1} \right)f = \delta (V)\,, \hfill \\ f(S) = 0\,, \hfill \\ f( - \infty ) = 0\,. \hfill \\ \end{gathered} $$

(16)

Equation (16) can be reduced to a tridiagonal linear simultaneous equation, which is quick and easy to solve (Press et al. 1992). On the other hand, from the fact that the relation ${p_{{0 \to S}}}(T) = \int {dVP(V,\theta ){p_{{V \to S}}}(T - \theta )}$ is valid when T > θ because Eq. (9) is a Markov process, the second term of Eq. (13) can be written as

$$ c\int_{{ - \infty }}^S {dVf(V)} \,\,\int_0^{\infty } {dt\,t{p_{{V \to S}}}(t)} $$

(17)

Here, $\,\int_0^{\infty } {dt\,t{p_{{V \to S}}}(t)}$ is the first moment of the first-passage-time distribution of Eq. (9) for λ _t = λ ⁽¹⁾. We denote it by ${t_1}(S|V)$. The explicit expression of ${t_1}(S|V)$ for the Ornstein-Uhlenbeck process has already been derived (Ricciardi and Sato 1988), and we can quickly compute it using expansion formulas (Ricciardi and Sato 1988; Keilson and Ross 1975; Shinomoto et al. 1999). Finally, from Eqs. (14) and (17), we obtain the following fast-to-compute form:

$$ R = 1 - \int_{{ - \infty }}^S {\left[ {1 - c{t_1}(S|V)} \right]f(V)dV} $$

(18)

Appendix C: Bayes-optimal detection algorithm

The Bayes-optimal change-point has been determined based on the method of Peskir and Shiryaev (2002). In this method, the evolution of the likelihood ratio φ _t is first computed with

$$ d{\phi_t} = (1 + {\phi_t})/\bar{\theta } dt + \left( {{\lambda^{{(1)}}}/{\lambda^{{(0)}}} - 1} \right) \,{\phi_{{t - }}} d({X_t} - {\lambda^{{(0)}}}t), $$

(19)

where ${\phi_{{t - }}} = \mathop{{\lim }}\limits_{{s \to t - 0}} {\phi_s}$. The algorithm detects a change underlying the input signal X _t when φ _t first reaches a stopping point, which has been determined so as to minimize the risk R given λ ⁽⁰⁾, λ ⁽¹⁾, $\overline \theta$ and c. According to the optimal stopping theory of Markov process (Shiryaev 1978) the stopping point is given by ${B_{ * }}/(1 - {B_{ * }})$, where B _* satisfies the following conditions:

$$ \begin{array}{*{20}{c}} {(LV)(\pi ) = - c\pi } \hfill & {(0 < \pi < {B_{ * }}),} \hfill \\ {V(\pi ) = 1 - \pi } \hfill & {({B_{ * }} \leqslant \pi \leqslant 1),} \hfill \\ {V({B_{ * }} - ) = 1 - {B_{ * }}} \hfill & {{\text{(continuous fit),}}} \hfill \\ \end{array} $$

(20)

where the differential-difference operator L was defined as

$$ \begin{gathered} (Lf)(\pi ) = \left( {\,1/\bar{\theta } - ({\lambda^{{(1)}}} - {\lambda^{{(0)}}})\pi } \right)\,\,(1 - \pi )\,f\prime(\pi ) \\ + \left( {{\lambda^{{(1)}}}\pi + {\lambda^{{(0)}}}(1 - \pi )} \right)\,\left( {f\left( {\frac{{{\lambda^{{(1)}}}\pi }}{{{\lambda^{{(1)}}}\pi + {\lambda^{{(0)}}}(1 - \pi )}}} \right) - f(\pi )} \right). \\ \end{gathered} $$

(21)

This free-boundary differential-difference problem is solved using the algorithm in (Peskir and Shiryaev 2002).