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Dendritic Pooling of Noisy Threshold Processes Can Explain Many Properties of a Collision-Sensitive Visual Neuron.

Keil MS - PLoS Comput. Biol. (2015)

Bottom Line: Then, an approximative power law is obtained as a result of pooling these channels.We show that with this mechanism one can successfully predict many response characteristics of the Lobula Giant Movement Detector Neuron (LGMD).Moreover, the results depend critically on noise in the inhibitory pathway, but they are fairly robust against noise in the excitatory pathway.

View Article: PubMed Central - PubMed

Affiliation: Basic Psychology Department, University of Barcelona, Barcelona, Spain; Institute of Brain, Bahaviour and Cognition (I3C), University of Barcelona, Barcelona, Spain.

ABSTRACT
Power laws describe brain functions at many levels (from biophysics to psychophysics). It is therefore possible that they are generated by similar underlying mechanisms. Previously, the response properties of a collision-sensitive neuron were reproduced by a model which used a power law for scaling its inhibitory input. A common characteristic of such neurons is that they integrate information across a large part of the visual field. Here we present a biophysically plausible model of collision-sensitive neurons with η-like response properties, in which we assume that each information channel is noisy and has a response threshold. Then, an approximative power law is obtained as a result of pooling these channels. We show that with this mechanism one can successfully predict many response characteristics of the Lobula Giant Movement Detector Neuron (LGMD). Moreover, the results depend critically on noise in the inhibitory pathway, but they are fairly robust against noise in the excitatory pathway.

No MeSH data available.


Logarithmic encoding of η(t).(a)The logarithmically encoded η-function  is “approximately exponentiated” by a power law with exponent p according to gp(t) = [logη(t)]pEq (6). The corresponding curves of gp (with p = 1,2,3,4,5 for l/v = 30ms) are denoted in the legend. For control, the sum  (exponential series: dash-dotted curve), and the η-function without log-encoding (gray curve) are also shown. Note that the curves become flatter with increasing p (and also with increasing l/v—not shown), what could make the detection of the maxima more uncertain in the presence of noise (cf. Fig 2b). (b)Applying a sigmoidal function 𝒮(x) = [1 + exp(− ax)]−1 (with a = 1) to the properly normalized curves gp(t) (with p = 3) produces curves which resemble true LGMD responses Eq (8). The curves shown here correspond to different halfsize-to-velocity ratios l/v = 10,20,40,40,50ms (see legend). The peak amplitude of the curves decreases with increasing halfsize-to-velocity ratio, but also with increasing a and p, respectively (not shown). (c) Identical with figure panel b, but here all curves 𝒮(g3) were re-scaled to the range from 0 to 1. For comparison, the corresponding η-functions are drawn alongside with dashed lines (α = 4.7, δ = 0, and tc = 0.5s in all figure panels).
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pcbi.1004479.g001: Logarithmic encoding of η(t).(a)The logarithmically encoded η-function is “approximately exponentiated” by a power law with exponent p according to gp(t) = [logη(t)]pEq (6). The corresponding curves of gp (with p = 1,2,3,4,5 for l/v = 30ms) are denoted in the legend. For control, the sum (exponential series: dash-dotted curve), and the η-function without log-encoding (gray curve) are also shown. Note that the curves become flatter with increasing p (and also with increasing l/v—not shown), what could make the detection of the maxima more uncertain in the presence of noise (cf. Fig 2b). (b)Applying a sigmoidal function 𝒮(x) = [1 + exp(− ax)]−1 (with a = 1) to the properly normalized curves gp(t) (with p = 3) produces curves which resemble true LGMD responses Eq (8). The curves shown here correspond to different halfsize-to-velocity ratios l/v = 10,20,40,40,50ms (see legend). The peak amplitude of the curves decreases with increasing halfsize-to-velocity ratio, but also with increasing a and p, respectively (not shown). (c) Identical with figure panel b, but here all curves 𝒮(g3) were re-scaled to the range from 0 to 1. For comparison, the corresponding η-functions are drawn alongside with dashed lines (α = 4.7, δ = 0, and tc = 0.5s in all figure panels).

Mentions: If the exponent p is fixed, then the biophysical implementation of o + g1, 3, 5, … and o − g2, 4, 6, …, respectively, should be feasible. Nevertheless, a general drawback of “undoing” the logarithmic transformation with a power law would be that the maximum of the LGMD membrane potential is strongly flattened: The bigger p is, the more (Fig 1a). As a consequence, the reliability of detecting the time of peak firing drops already for small noise amplitudes (Fig 2b). The different phases of escape jumps seem to occur around the peak firing rate of the DCMD [25], although this does not automatically imply that the peak time is explicitly decoded by the motor system of the locust. Experimentally measured timings of DCMD peak firing have standard deviations of ≈ 50ms [20]. The comparatively high variability of the response maximum as predicted by g3(t) (Fig 2b) stands in contrast to these observations, since standard deviations are higher than 100ms even for small noise levels. It seems therefore that the sole power law of Eq (6) cannot reproduce LGMD firing patterns. Is it possible to regain the undistorted, “LGMD-like” response curves from gp(t)? Reference [17] holds a general mathematical solution in order to map the η-function to the firing rate of the LGMD. The mapping function is a (here simplified) sigmoid 𝒮(x) = [1 + exp(− ax)]−1. The value of a determines whether the activation from zero to one proceeds in a more linear way (a ≪ 1) or according to step-function (a ≫ 1). With gp = (logη)pEq (6) one obtains𝒮(gp)=ηa(logη)p-11+ηa(logη)p-1(8)and specifically 𝒮(g1) = ηa/(1+ηa). With an appropriate choice of the parameters a and p, respectively, the function 𝒮(gp) reduces the initial activity compared to the ordinary η-function. In this way the peak is more pronounced (see Fig 1c). However, with increasing p and a, respectively, the peak amplitudes of curves with higher l/v decrease relative to low halfsize-to-velocity ratios (see Fig 1b). The peak values for a set of l/v values could quickly span several orders of magnitude for already “moderate values” a > 1 (e.g. a = 5) and p > = 1.


Dendritic Pooling of Noisy Threshold Processes Can Explain Many Properties of a Collision-Sensitive Visual Neuron.

Keil MS - PLoS Comput. Biol. (2015)

Logarithmic encoding of η(t).(a)The logarithmically encoded η-function  is “approximately exponentiated” by a power law with exponent p according to gp(t) = [logη(t)]pEq (6). The corresponding curves of gp (with p = 1,2,3,4,5 for l/v = 30ms) are denoted in the legend. For control, the sum  (exponential series: dash-dotted curve), and the η-function without log-encoding (gray curve) are also shown. Note that the curves become flatter with increasing p (and also with increasing l/v—not shown), what could make the detection of the maxima more uncertain in the presence of noise (cf. Fig 2b). (b)Applying a sigmoidal function 𝒮(x) = [1 + exp(− ax)]−1 (with a = 1) to the properly normalized curves gp(t) (with p = 3) produces curves which resemble true LGMD responses Eq (8). The curves shown here correspond to different halfsize-to-velocity ratios l/v = 10,20,40,40,50ms (see legend). The peak amplitude of the curves decreases with increasing halfsize-to-velocity ratio, but also with increasing a and p, respectively (not shown). (c) Identical with figure panel b, but here all curves 𝒮(g3) were re-scaled to the range from 0 to 1. For comparison, the corresponding η-functions are drawn alongside with dashed lines (α = 4.7, δ = 0, and tc = 0.5s in all figure panels).
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Related In: Results  -  Collection

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pcbi.1004479.g001: Logarithmic encoding of η(t).(a)The logarithmically encoded η-function is “approximately exponentiated” by a power law with exponent p according to gp(t) = [logη(t)]pEq (6). The corresponding curves of gp (with p = 1,2,3,4,5 for l/v = 30ms) are denoted in the legend. For control, the sum (exponential series: dash-dotted curve), and the η-function without log-encoding (gray curve) are also shown. Note that the curves become flatter with increasing p (and also with increasing l/v—not shown), what could make the detection of the maxima more uncertain in the presence of noise (cf. Fig 2b). (b)Applying a sigmoidal function 𝒮(x) = [1 + exp(− ax)]−1 (with a = 1) to the properly normalized curves gp(t) (with p = 3) produces curves which resemble true LGMD responses Eq (8). The curves shown here correspond to different halfsize-to-velocity ratios l/v = 10,20,40,40,50ms (see legend). The peak amplitude of the curves decreases with increasing halfsize-to-velocity ratio, but also with increasing a and p, respectively (not shown). (c) Identical with figure panel b, but here all curves 𝒮(g3) were re-scaled to the range from 0 to 1. For comparison, the corresponding η-functions are drawn alongside with dashed lines (α = 4.7, δ = 0, and tc = 0.5s in all figure panels).
Mentions: If the exponent p is fixed, then the biophysical implementation of o + g1, 3, 5, … and o − g2, 4, 6, …, respectively, should be feasible. Nevertheless, a general drawback of “undoing” the logarithmic transformation with a power law would be that the maximum of the LGMD membrane potential is strongly flattened: The bigger p is, the more (Fig 1a). As a consequence, the reliability of detecting the time of peak firing drops already for small noise amplitudes (Fig 2b). The different phases of escape jumps seem to occur around the peak firing rate of the DCMD [25], although this does not automatically imply that the peak time is explicitly decoded by the motor system of the locust. Experimentally measured timings of DCMD peak firing have standard deviations of ≈ 50ms [20]. The comparatively high variability of the response maximum as predicted by g3(t) (Fig 2b) stands in contrast to these observations, since standard deviations are higher than 100ms even for small noise levels. It seems therefore that the sole power law of Eq (6) cannot reproduce LGMD firing patterns. Is it possible to regain the undistorted, “LGMD-like” response curves from gp(t)? Reference [17] holds a general mathematical solution in order to map the η-function to the firing rate of the LGMD. The mapping function is a (here simplified) sigmoid 𝒮(x) = [1 + exp(− ax)]−1. The value of a determines whether the activation from zero to one proceeds in a more linear way (a ≪ 1) or according to step-function (a ≫ 1). With gp = (logη)pEq (6) one obtains𝒮(gp)=ηa(logη)p-11+ηa(logη)p-1(8)and specifically 𝒮(g1) = ηa/(1+ηa). With an appropriate choice of the parameters a and p, respectively, the function 𝒮(gp) reduces the initial activity compared to the ordinary η-function. In this way the peak is more pronounced (see Fig 1c). However, with increasing p and a, respectively, the peak amplitudes of curves with higher l/v decrease relative to low halfsize-to-velocity ratios (see Fig 1b). The peak values for a set of l/v values could quickly span several orders of magnitude for already “moderate values” a > 1 (e.g. a = 5) and p > = 1.

Bottom Line: Then, an approximative power law is obtained as a result of pooling these channels.We show that with this mechanism one can successfully predict many response characteristics of the Lobula Giant Movement Detector Neuron (LGMD).Moreover, the results depend critically on noise in the inhibitory pathway, but they are fairly robust against noise in the excitatory pathway.

View Article: PubMed Central - PubMed

Affiliation: Basic Psychology Department, University of Barcelona, Barcelona, Spain; Institute of Brain, Bahaviour and Cognition (I3C), University of Barcelona, Barcelona, Spain.

ABSTRACT
Power laws describe brain functions at many levels (from biophysics to psychophysics). It is therefore possible that they are generated by similar underlying mechanisms. Previously, the response properties of a collision-sensitive neuron were reproduced by a model which used a power law for scaling its inhibitory input. A common characteristic of such neurons is that they integrate information across a large part of the visual field. Here we present a biophysically plausible model of collision-sensitive neurons with η-like response properties, in which we assume that each information channel is noisy and has a response threshold. Then, an approximative power law is obtained as a result of pooling these channels. We show that with this mechanism one can successfully predict many response characteristics of the Lobula Giant Movement Detector Neuron (LGMD). Moreover, the results depend critically on noise in the inhibitory pathway, but they are fairly robust against noise in the excitatory pathway.

No MeSH data available.