Phase analysis method for burst onset prediction

The response of bursting neurons to fluctuating inputs is usually hard to predict, due to their strong nonlinearity. For the same reason, decoding the injected stimulus from the activity of a bursting neuron is generally difficult. In this paper we propose a method describing (for neuron models) a mechanism of phase coding relating the burst onsets with the phase profile of the input current. This relation suggests that burst onset may provide a way for postsynaptic neurons to track the input phase. Moreover, we define a method of phase decoding to solve the inverse problem and estimate the likelihood of burst onset given the input state. Both methods are presented here in a unified framework, describing a complete coding-decoding procedure. This procedure is tested by using different neuron models, stimulated with different inputs (stochastic, sinusoidal, up, and down states). The results obtained show the efficacy and broad range of application of the proposed methods. Possible applications range from the study of sensory information processing, in which phase-of-firing codes are known to play a crucial role, to clinical applications such as deep brain stimulation, helping to design stimuli in order to trigger or prevent neural bursting.

Figure 1

Block scheme describing at a glance the phase coding part of the method applied to obtain raster plots of the phase profiles. The input signal $x\left(t\right)$ is filtered to obtain the current $I\left(t\right)$, which is the input of both a two-compartment neuron model with active currents (reduced Traub model) (see Appendix pp1 and [21]) and a phase detection block (see Appendix pp4). From the neuron output $V\left(t\right)$ we extract the spike times and then the IBIs. By combining this information with the phase detected from $I\left(t\right)$, we obtain the phase profiles corresponding to IBIs. Finally, these phase profiles are ordered according to their lengths [the vector $\tilde{\mathcal{L}}$ is defined in Eq. (2)] and represented in a raster plot.

Figure 2

(a)–(d) Signals $x\left(t\right)$ (light gray lines) and corresponding (after filtering in the frequency band $[3,7]\mathrm{Hz})$ synaptic input currents $I\left(t\right)$ (black lines) in nA for white, pink, Ornstein, and brown noise, respectively. (e)–(h) PSDs (in dB) of the corresponding stochastic processes, both without filtering (light gray lines) and with filtering in the band $[3,7]\mathrm{Hz}$ (black lines). Since the ordinate axis is common to each row, it is reported only in the leftmost figure.

Figure 3

Detail of the (a) input stimulus $I\left(t\right)$ (white noise filtered in the frequency band $[3,7]\mathrm{Hz})$ and (b) corresponding phase ${\phi }_I\left(t\right)$ and (c) membrane potential $V\left(t\right)$. The gray closed (open) circles mark the starting (final) point of each IBI. (d) Raster plot of phase profiles ${\phi }_k\left(\tau \right)$, obtained from a simulation of 1000 s. The colors (gray levels) code the phase amplitudes.

Figure 4

Raster plots of phase profiles ${\phi }_k\left(\tau \right)$, obtained from a simulation of 1000 s with (a) brown noise, (b) Ornstein noise, and (c) pink noise filtered in the frequency band $[3,7]\mathrm{Hz}$. Since the $k$ axis is common to each row, it is shown only in the leftmost figure.

Figure 5

(a) Raster plot of the phase profiles ${\phi }_k$ generated by a stochastic stimulation (comprising all four kinds of stimuli considered) filtered in the band $[3,7]\mathrm{Hz}$ and corresponding raster plots of (b) ${\mathring{\mu }}_l\left(\tau \right)$ (map $\mathrm{\Psi })$ and (c) ${\mathring{\sigma }}_l\left(\tau \right)$ (map ${\mathrm{\Psi }}_{\sigma })$. Since the $k$ axis is common to each row, it is shown only in the leftmost figure.

Figure 6

Histograms of the IBI lengths contained within the subintervals (a) ${\lambda }_{l_A}$, (b) ${\lambda }_{l_B}$, and (c) ${\lambda }_{l_C}$ [vertical transparent green (light gray) rectangles in Fig. 5]. The mean values ${\mu }_{l_A},{\mu }_{l_B}$, and ${\mu }_{l_C}$ are marked by vertical red (gray) lines. Also shown is the zoom of the raster plots of phase profiles ${\phi }_k\left(t\right)$ belonging to (d) ${\varphi }_{l_A}$ $\left(k\in [1144,2976]\right)$, (e) ${\varphi }_{l_B}$ $\left(k\in [8524,9955]\right)$, and (f) ${\varphi }_{l_C}$ $\left(k\in [15198,17551]\right)$, highlighted with horizontal transparent blue (gray) rectangles in Fig. 5. The corresponding PDFs (g) ${\mathring{f}}_{l_A}(\phi ,\tau )$, (h) ${\mathring{f}}_{l_B}(\phi ,\tau )$, and (i) ${\mathring{f}}_{l_C}(\phi ,\tau )$ are shown in logarithmic color (gray level) scale. Also shown are the corresponding circular mean phase profiles (black curves) (j) ${\mathring{\mu }}_{l_A}\left(\tau \right)$, (k) ${\mathring{\mu }}_{l_B}\left(\tau \right)$, and (l) ${\mathring{\mu }}_{l_C}\left(\tau \right)$ with confidence intervals (gray regions).

Figure 7

(a) Raster plot of $r_l\left(\tau \right)$ corresponding to the phase profiles ${\phi }_k\left(\tau \right)$ shown in Fig. 5; the black vertical segments mark two lengths $l_a$ and $l_b$. (b) Circular mean profiles of the phases corresponding to $l_a$ (gray curve) and $l_b$ (black curve).

Figure 8

Comparison of different kinds of raster plots (columns) for different frequency bands (rows). Raster plots of phase profiles ${\phi }_k\left(\tau \right)$ (first column), circular mean phase profile ${\mathring{\mu }}_l\left(\tau \right)$ (second column), circular standard deviation profile ${\mathring{\sigma }}_l\left(\tau \right)$ (third column), and phase-conditional firing probability $r_l\left(\tau \right)$ (fourth column) extracted from the neural response to stochastic stimulation with all four kinds of synaptic stimuli considered, filtered in the bands $[1,5]\mathrm{Hz}$ (first row), $[5,9]\mathrm{Hz}$ (second row), $[9,13]\mathrm{Hz}$ (third row), $[13,17]\mathrm{Hz}$ (fourth row), and $[17,21]\mathrm{Hz}$ (last row). Since the $\tau$ axis (in ms) is common to all figures and the $k$ axis is common to each row of figures, we show them only in the leftmost and bottom figures. The other figures keep only the axis ticks, as reference.

Figure 9

Raster plot of ${\phi }_k\left(\tau \right)$ (first column), ${\mathring{\mu }}_l\left(\tau \right)$ (second column), ${\mathring{\sigma }}_l\left(\tau \right)$ (third column), and $r_l\left(\tau \right)$ (fourth column) extracted from the neural response to stochastic stimulation obtained by using all four kinds of synaptic stimuli considered, filtered in the bands $[3,7]\mathrm{Hz}$ for the pyramidal (first row), adaptive exponential integrate and fire (second row), and Izhikevich (third row) neuron models. See Fig. 8 caption about the axes.

Figure 10

Detail of the neural response [membrane potential $V\left(t\right)]$ to (a) the synaptic input $I\left(t\right)$, corresponding to (b) the phase profile ${\phi }_I\left(t\right)$. (c) Phase profiles ${\phi }_k$ [zoom of ${\phi }_I\left(t\right)]$ and ${\tilde{\phi }}_L\left(\tau \right)=\mathrm{\Psi }\left[L\right]$. (d) Phase conditional time-dependent burst-onset probability $r_l\left(\tau \right)$.

Figure 11

(a) Detail of the input stimulus $I\left(t\right)$ (gray line, right axis) and corresponding membrane potential $V\left(t\right)$ (black line, left axis). Also shown are the raster plots of the phase profiles (b) ${\phi }_k$, (c) ${\mathring{\mu }}_l\left(\tau \right)$, (d) ${\mathring{\sigma }}_l\left(\tau \right)$, and (e) $r_l\left(\tau \right)$ obtained from the complete stimulation (of length 1000 s). Since the $k$ axis is common to (b)–(e), it is shown only in the leftmost figure.