Comparison of local measures of spike time irregularity and relating variability to firing rate in motor cortical neurons

Abstract

Spike time irregularity can be measured by the coefficient of variation. However, it overestimates the irregularity in the case of pronounced firing rate changes. Several alternative measures that are local in time and therefore relatively rate-independent were proposed. Here we compared four such measures: CV₂, LV, IR and SI. First, we asked which measure is the most efficient for time-resolved analyses of experimental data. Analytical results show that CV₂ has the less variable estimates. Second, we derived useful properties of CV₂ for gamma processes. Third, we applied CV₂ on recordings from the motor cortex of a monkey performing a delayed motor task to characterize the irregularity, that can be modulated or not, and decoupled or not from firing rate. Neurons with a CV₂-rate decoupling have a rather constant CV₂ and discharge mainly irregularly. Neurons with a CV₂-rate coupling can modulate their CV₂ and explore a larger range of CV₂ values.

Appendices

Appendix A: Probability Density of m

In the following we provide a general way to calculate the probability density function (PDF) of m for any of the four local measures for a gamma-process with rate r and shape parameter α.

Let m(τ,τ′) be the individual value of the measure which is a function of the two consecutive ISIs τ and τ′. The cumulative distribution of m describes the probability of finding m larger than a given value m ₀:

$$ C\left( {m_0 } \right) = P\left[ {m > m_0 } \right] $$

To calculate this function, we define an auxiliary variable Φ(m) for getting:

$$ m > m_0 \Rightarrow \tau \prime\kern1.5pt<\kern1.5pt\tau .\Phi \left( {m_0 } \right) $$

Hence, the cumulative function can be expressed as the probability of finding an ISI less than τΦ(m ₀) integrated over all possible values of τ:

$$ P\left[ {m > m_0 } \right] = \int\limits_0^{\infty } {P\left[ {\tau \prime\kern1.5pt<\kern1.5pt\tau \Phi \left( {m_0 } \right)} \right]\rho_{\alpha } \left( \tau \right)d\tau } $$

where ρ _α (τ) is the gamma distribution (Eq. (9)). Note that for a gamma-process, the probability of finding an ISI less than T is:

$$ \int\limits_0^T {\rho_{\alpha } \left( \tau \right)d\tau = \frac{1}{{\Gamma \left( \alpha \right)}}\int\limits_0^{{r\alpha T}} {u^{{\alpha - 1}} e^{- u} du} } $$

Thus,

$$ P\left[ {m > m_0 } \right] = \int\limits_0^{\infty } {\left\{ {\frac{1}{{\Gamma \left( \alpha \right)}}\int\limits_0^{{r\alpha \tau \Phi \left( {m_0 } \right)}} {u^{{\alpha - 1}} e^{- u} du} } \right\}\frac{{\left( {\alpha r} \right)^{\alpha } }}{{\Gamma \left( \alpha \right)}}\tau^{{\alpha - 1}} e^{{ - r\alpha \tau }} d\tau } $$

Finally, the PDF is the derivative of the cumulative, and thus:

$$ f_{\alpha } (m) = \frac{dP}{{dm_0 }} = \frac{{\Gamma \left( {2\alpha } \right)}}{{\Gamma \left( \alpha \right)^2 }}\frac{{\Phi \prime \left( {m_0 } \right)\Phi \left( {m_0 } \right)^{{\alpha - 1}} }}{{\left[ {1 + \Phi \left( {m_0 } \right)} \right]^{{2\alpha }} }} $$

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The PDF f _α (m) gives the probability of finding m in the interval [m, m + dm] and depends only on the shape parameter α.

Remark: for Φ = Identity we obtain the PDF for the ratio of two consecutive ISIs, that is:

$$ f_{\alpha } \left( {x = \frac{{\tau \prime }}{\tau }} \right) = \frac{{\Gamma \left( {2\alpha } \right)}}{{\Gamma \left( \alpha \right)^2 }}\frac{{x^{{\alpha - 1}} }}{{\left[ {1 + x} \right]^{{2\alpha }} }} $$

This function peaks at (α-1)/(α + 1) for α > 1.

Appendix B: Probability Density and expected value of CV₂

Now we explicitly express the PDF of $m^{{CV_2 }}$ and its expected value. For this we define the auxiliary variable:

$$ X = 2\frac{{\tau - \tau \prime }}{{\tau + \tau \prime }} \Rightarrow m = \left| X \right| $$

X is bounded between −2 and 2 and m is bound between 0 and 2. The cumulative function of X is:

$$ P\left[ {X > X_0 } \right] = \int\limits_0^{\infty } P \left[ {\tau \prime\kern1.5pt<\kern1.5pt\tau \frac{{2 - X_0 }}{{2 + X_0 }}} \right]\rho_{\alpha } \left( \tau \right)d\tau $$

Using Eq. (14) with Φ(X) = (2-X)/(2 + X) we find:

$$ f_{\alpha } (X) = - \frac{1}{{4^{{2\alpha - 1}} }}\frac{{\Gamma \left( {2\alpha } \right)}}{{\Gamma \left( \alpha \right)^2 }}\left( {4 - X^2 } \right)^{{\alpha - 1}} $$

This function is symmetric and, thus, the PDF of m is:

$$ f_{\alpha } (m) = 2\left| {f_{\alpha } (X)} \right| = \frac{2}{{4^{{2\alpha - 1}} }}\frac{{\Gamma \left( {2\alpha } \right)}}{{\Gamma \left( \alpha \right)^2 }}\left( {4 - m^2 } \right)^{{\alpha - 1}} $$

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Note that for a Poisson process (α = 1) the probability density of m is flat and equal to 1/2. The expected value of CV₂ can be expressed by means of the PDF:

$$ CV_2 \left( \alpha \right) = \int\limits_0^2 {mf_{\alpha } (m)dm = \frac{{4^{{ - \alpha + 1}} }}{\alpha }} \frac{{\Gamma \left( {2\alpha } \right)}}{{\Gamma \left( \alpha \right)^2 }} $$

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The same procedure can be applied to calculate the PDF of LV, IR, and SI using Eq. (14) and noting that the PDF of an auxiliary variable X, g _α (X), and the PDF of m, f _α (m), are related by:

$$ f_{\alpha } (m) = \frac{dP}{dm} = g_{\alpha } (X)\left| {\frac{{\partial X}}{{\partial m}}} \right| $$

We propose the following auxiliary variables and expressions of the PDF of LV, IR, and SI:

$$ \begin{array}{*{20}c} {{\text{LV:}}\;\,X = 3\frac{{\tau - \tau \prime }}{{\tau + \tau \prime }} \Rightarrow m=\left| X \right|^2; } & {{\text{PDF:}}\,\;f_{\alpha } (m)=\frac{{\Gamma \left( {2\alpha } \right)}}{{\Gamma \left( \alpha \right)^2 }}\left( {\frac{1}{{2\sqrt 3 }}} \right)^{{2\alpha - 1}} \frac{{\left( {3 - m} \right)^{{\alpha - 1}} }}{{\sqrt m }}} \end{array} $$

$$ \begin{array}{*{20}c} {{\text{IR:}}\,\;X = \frac{\tau }{{\tau \prime }} \Rightarrow m\left| {\log (X)} \right|;} & {{\text{PDF:}}\,\;f_{\alpha } (m) = \frac{{\Gamma \left( {2\alpha } \right)}}{{\Gamma \left( \alpha \right)^2 }}\frac{{2e^{{\alpha m}} }}{{\left( {1 + e^m } \right)^{{2\alpha }} }}} \end{array} $$

$$ \begin{array}{*{20}c} {{\text{SI:}}\,\;X = \frac{{\left( {\tau + \tau \prime } \right)^2 }}{{4\tau \tau \prime }} \Rightarrow m=\frac{1}{2}\log (X);} & {{\text{PDF:}}\,\;f_{\alpha } (m) = 4\frac{{\Gamma \left( {2\alpha } \right)}}{{\Gamma \left( \alpha \right)^2 }}\frac{{Q\prime \left( {e^{2m} } \right)Q\left( {e^{2m} } \right)^{{\alpha - 1}} }}{{\left( {1 + Q\left( {e^{2m} } \right)} \right)^{{2\alpha }} }}e^{2m} } \end{array} $$

where: $Q(x) = - 1 + 2x + 2\sqrt {x^2 - x}$.

Appendix C: Effect of an absolute refractory period

We now explore how the incorporation of an absolute refractory period influences the properties of CV₂. We extend the Poisson model to account for refractoriness by setting the instantaneous firing rate to zero for a time ε immediately after a spike is fired. After this dead-time the firing rate returns immediately to the value in absence of refractoriness. In this model the interval distribution is:

$$ \rho \left( \tau \right) = \left\{ {\begin{array}{*{20}c} 0 \hfill & {{\text{if}}\,\;\tau\kern1.5pt<\kern1.5pt\varepsilon } \hfill \\ {re^{{ - r\left( {\tau - \varepsilon } \right)}} } \hfill & {\text{otherwise}} \hfill \\ \end{array} } \right. $$

To calculate the probability density of m we use the same strategy as before. First we calculate the cumulative function of X (with |X| = m):

$$ P\left[ {X > X_0 } \right] = \int {_{{\tau \frac{{2 - X_0 }}{{2 + X_0 }} > \varepsilon }} P\left[ {\tau \prime\kern1.5pt<\kern1.5pt\tau \frac{{2 - X_0 }}{{2 + X_0 }}} \right]\rho_{\alpha } \left( \tau \right)d\tau } $$

Two cases must be separated to satisfy the condition τ > ε: if X ₀ > 0 the integration has to be done from ε(2-X ₀)/(2 + X ₀) to ∞, whereas if X ₀ < 0 the integration has to be done from ε to ∞. Taking the derivative of this cumulative function gives the probability density of X:

$$ f\left( {X_0 } \right) = \frac{dP}{dX} = \left\{ {\begin{array}{*{20}c} {\left[ {\frac{1}{4} + \frac{{r\varepsilon }}{{2 - X_0 }}} \right]\exp \left( { - r\varepsilon \frac{{2X_0 }}{{2 - X_0 }}} \right)} \hfill & {{\text{if}}\,\;0 \le X_0\kern1.5pt<\kern1.5pt2} \hfill \\ { - \left[ {\frac{1}{4} + \frac{{r\varepsilon }}{{2 + X_0 }}} \right]\exp \left( {r\varepsilon \frac{{2X_0 }}{{2 + X_0 }}} \right)} \hfill & {{\text{if}}\,\; - 2\kern1.5pt<\kern1.5ptX_0\kern1.5pt<\kern1.5pt0} \hfill \\ \end{array} } \right. $$

Hence, the probability density of m is:

$$ f(m) = 2\left[ {\frac{1}{4} + \frac{{r\varepsilon }}{2 - m}} \right]\exp \left( { - r\varepsilon \frac{2m}{2 - m}} \right) $$

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Note that the probability density of m in the case of refractoriness is not flat as originally proposed by Holt et al. (1996) and it now depends on rε, i.e. the fraction of time spend in refractory period.

In the same way, we can calculate the probability density function of m for a gamma process with an absolute refractory period ε, that is given by:

$$ f_{\alpha } (m) = \frac{{8e^{{ - r\varepsilon \alpha \frac{2m}{2 - m}}} }}{{\Gamma \left( \alpha \right)^2 \left( {2 - m} \right)^{{\alpha + 1}} }}\int\limits_0^{\infty } {\left( {u - r\varepsilon \alpha } \right)\left[ {u^2 \left( {2 + m} \right) + u\left( {2mr\varepsilon \alpha } \right)} \right]^{{\alpha - 1}} e^{{ - u\frac{4}{2 - m}}} du} $$

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Finally, the expected value of CV₂ for a gamma process with absolute refractory period ε is:

$$ CV_2 = \int\limits_0^2 {mf_{\alpha } (m)dm} $$

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