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Reconstructing stimuli from the spike times of leaky integrate and fire neurons.

Gerwinn S, Macke JH, Bethge M - Front Neurosci (2011)

Bottom Line: Reconstructing stimuli from the spike trains of neurons is an important approach for understanding the neural code.One of the difficulties associated with this task is that signals which are varying continuously in time are encoded into sequences of discrete events or spikes.For the special case of spike trains generated by leaky integrate and fire neurons, noise can be introduced by allowing variations in the threshold every time a spike is released.

View Article: PubMed Central - PubMed

Affiliation: Werner Reichardt Center for Integrative Neuroscience, University of Tübingen Tübingen, Germany.

ABSTRACT
Reconstructing stimuli from the spike trains of neurons is an important approach for understanding the neural code. One of the difficulties associated with this task is that signals which are varying continuously in time are encoded into sequences of discrete events or spikes. An important problem is to determine how much information about the continuously varying stimulus can be extracted from the time-points at which spikes were observed, especially if these time-points are subject to some sort of randomness. For the special case of spike trains generated by leaky integrate and fire neurons, noise can be introduced by allowing variations in the threshold every time a spike is released. A simple decoding algorithm previously derived for the noiseless case can be extended to the stochastic case, but turns out to be biased. Here, we review a solution to this problem, by presenting a simple yet efficient algorithm which greatly reduces the bias, and therefore leads to better decoding performance in the stochastic case.

No MeSH data available.


Related in: MedlinePlus

Spikes imply linear constraints on the stimulus. (A) Equality constraints only: The stimulus (blue) consists of a superposition of two basis function (one sine and one cosine function). The resulting membrane potential is plotted in black. Whenever the potential reaches the threshold (gray) a spike is fired. The linear constraint originating from the equality constraints of the first interspike interval are plotted in green in the upper plot. A discrete set of points along this linear constraint are selected and the corresponding membrane potentials are plotted in green in the lower plot. The same is shown for the second interspike interval in red. The perfect reconstruction of the stimulus (coefficients) is found by the intersection of the linear constraints. The brightness of the background indicates the probability of each pair of coefficients according to the prior distribution. (B) Equality and inequality constraints: The same configuration as in (A) is plotted, but additionally the inequality constraints imposed by the period of silence between spike times are taken into account as well. This results in a smaller subspace of possible solutions as shown in the upper plot.
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Figure 3: Spikes imply linear constraints on the stimulus. (A) Equality constraints only: The stimulus (blue) consists of a superposition of two basis function (one sine and one cosine function). The resulting membrane potential is plotted in black. Whenever the potential reaches the threshold (gray) a spike is fired. The linear constraint originating from the equality constraints of the first interspike interval are plotted in green in the upper plot. A discrete set of points along this linear constraint are selected and the corresponding membrane potentials are plotted in green in the lower plot. The same is shown for the second interspike interval in red. The perfect reconstruction of the stimulus (coefficients) is found by the intersection of the linear constraints. The brightness of the background indicates the probability of each pair of coefficients according to the prior distribution. (B) Equality and inequality constraints: The same configuration as in (A) is plotted, but additionally the inequality constraints imposed by the period of silence between spike times are taken into account as well. This results in a smaller subspace of possible solutions as shown in the upper plot.

Mentions: For the case of only two basis functions we have illustrated the constraints defined by two interspike intervals in Figure 3. For the first interspike interval the membrane potential has to fulfill the constraint of being at the reset potential at the beginning of the interval and at the threshold at the end of it. This restricts the possible stimuli to a linear subspace shown as a green line in the upper part of Figure 3A. A similar restriction results from the second interspike interval leading to the linear subspace plotted in red. The true coefficients c* that were used to generate the stimulus, therefore have to be at the intersection of these two constraint subspaces. In the case of underdetermined, i.e., uncertain stimulus reconstruction, there are more constraints which could be exploited. As can be seen in Figure 3, there are some coefficients which fulfill the threshold constraint but result in membrane potentials which cross the threshold between spikes. Incorporating these constraints into a decoding procedure is computationally more demanding, as in each time bin they only define an inequality constraint. Solving this would result in a linear program with as many constraints as there are time bins, which would be computationally infeasible in almost all interesting cases. For the example in Figure 3, we plotted the space of solutions, which additionally account for the inequality constraints in subplot B. Neglecting these constraints, we need possibly more observations, but are still able to reconstruct the stimulus perfectly eventually: The linear constraints implied by the observed spikes will eventually “rule out” all stimuli which violate one of the inequality constraints.


Reconstructing stimuli from the spike times of leaky integrate and fire neurons.

Gerwinn S, Macke JH, Bethge M - Front Neurosci (2011)

Spikes imply linear constraints on the stimulus. (A) Equality constraints only: The stimulus (blue) consists of a superposition of two basis function (one sine and one cosine function). The resulting membrane potential is plotted in black. Whenever the potential reaches the threshold (gray) a spike is fired. The linear constraint originating from the equality constraints of the first interspike interval are plotted in green in the upper plot. A discrete set of points along this linear constraint are selected and the corresponding membrane potentials are plotted in green in the lower plot. The same is shown for the second interspike interval in red. The perfect reconstruction of the stimulus (coefficients) is found by the intersection of the linear constraints. The brightness of the background indicates the probability of each pair of coefficients according to the prior distribution. (B) Equality and inequality constraints: The same configuration as in (A) is plotted, but additionally the inequality constraints imposed by the period of silence between spike times are taken into account as well. This results in a smaller subspace of possible solutions as shown in the upper plot.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3046364&req=5

Figure 3: Spikes imply linear constraints on the stimulus. (A) Equality constraints only: The stimulus (blue) consists of a superposition of two basis function (one sine and one cosine function). The resulting membrane potential is plotted in black. Whenever the potential reaches the threshold (gray) a spike is fired. The linear constraint originating from the equality constraints of the first interspike interval are plotted in green in the upper plot. A discrete set of points along this linear constraint are selected and the corresponding membrane potentials are plotted in green in the lower plot. The same is shown for the second interspike interval in red. The perfect reconstruction of the stimulus (coefficients) is found by the intersection of the linear constraints. The brightness of the background indicates the probability of each pair of coefficients according to the prior distribution. (B) Equality and inequality constraints: The same configuration as in (A) is plotted, but additionally the inequality constraints imposed by the period of silence between spike times are taken into account as well. This results in a smaller subspace of possible solutions as shown in the upper plot.
Mentions: For the case of only two basis functions we have illustrated the constraints defined by two interspike intervals in Figure 3. For the first interspike interval the membrane potential has to fulfill the constraint of being at the reset potential at the beginning of the interval and at the threshold at the end of it. This restricts the possible stimuli to a linear subspace shown as a green line in the upper part of Figure 3A. A similar restriction results from the second interspike interval leading to the linear subspace plotted in red. The true coefficients c* that were used to generate the stimulus, therefore have to be at the intersection of these two constraint subspaces. In the case of underdetermined, i.e., uncertain stimulus reconstruction, there are more constraints which could be exploited. As can be seen in Figure 3, there are some coefficients which fulfill the threshold constraint but result in membrane potentials which cross the threshold between spikes. Incorporating these constraints into a decoding procedure is computationally more demanding, as in each time bin they only define an inequality constraint. Solving this would result in a linear program with as many constraints as there are time bins, which would be computationally infeasible in almost all interesting cases. For the example in Figure 3, we plotted the space of solutions, which additionally account for the inequality constraints in subplot B. Neglecting these constraints, we need possibly more observations, but are still able to reconstruct the stimulus perfectly eventually: The linear constraints implied by the observed spikes will eventually “rule out” all stimuli which violate one of the inequality constraints.

Bottom Line: Reconstructing stimuli from the spike trains of neurons is an important approach for understanding the neural code.One of the difficulties associated with this task is that signals which are varying continuously in time are encoded into sequences of discrete events or spikes.For the special case of spike trains generated by leaky integrate and fire neurons, noise can be introduced by allowing variations in the threshold every time a spike is released.

View Article: PubMed Central - PubMed

Affiliation: Werner Reichardt Center for Integrative Neuroscience, University of Tübingen Tübingen, Germany.

ABSTRACT
Reconstructing stimuli from the spike trains of neurons is an important approach for understanding the neural code. One of the difficulties associated with this task is that signals which are varying continuously in time are encoded into sequences of discrete events or spikes. An important problem is to determine how much information about the continuously varying stimulus can be extracted from the time-points at which spikes were observed, especially if these time-points are subject to some sort of randomness. For the special case of spike trains generated by leaky integrate and fire neurons, noise can be introduced by allowing variations in the threshold every time a spike is released. A simple decoding algorithm previously derived for the noiseless case can be extended to the stochastic case, but turns out to be biased. Here, we review a solution to this problem, by presenting a simple yet efficient algorithm which greatly reduces the bias, and therefore leads to better decoding performance in the stochastic case.

No MeSH data available.


Related in: MedlinePlus