A brief demonstration on how do early visual system cope with neural time delay problem by computational model

1. How Do Retina Cope With Time Delay?
Chuan Yu Hu
PI: Dr. Chi-Keung Chan
Institute of Physics, Academia Sinica
2020.8.24
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2. Introduction
Stimulus and Analysis method
Retina feedback model
Simulation result
Conclusion
Acknowledgement
Reference
Appendix
Outline
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3. Introduction | Neural signal has delay effect
Retina inner structure illustration
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• There is a time delay in the visual system during the neuron transmission
• In order to compensate with this delay, a retina need to predict the future
Schnapf, J. L., Kraft, T. W., & Baylor, D. A. (1987).
Inner Retina
Outer Retina
(RGC)
1. Time Delay (30-100ms )
2. Horizontal feedback
3. Parallel processing

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Stimulus and Analysis method
Time-lag mutual information – MI(𝜹𝒕)
Response is “predictive” if the time lag of
the peak of MI is positive.
𝑀𝐼(𝛿𝑡𝑝𝑒𝑎𝑘)
r(t)
S(t’)
Future
Past
S(t+Δ𝑡)
r(t)
Light intensity
𝑥𝑡+Δ𝑡 = (1 − Δ𝑡/𝜏)𝑥𝑡+𝜉𝑡 𝐷Δ𝑡
Ornstein-Uhlenbeck process (OU)
MI
(
)

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Retina feedback model
• Researchers have found different effects by controlling feedback within cone cell – horizontal cell
• Derived from experimental result, they have also built models for retina system
Drinnenberg, A. et al. How diverse retinal functions arise from feedback at the first visual synapse. Neuron 99, 117–134.e11 (2018).
Bipolar
Photoreceptor
Horizontal
cell
𝑥(𝑡)
𝑦(𝑡)
𝑧(𝑡)
Schematic of the model

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Simulation result of Retina feedback model
Drinnenberg, A. et al. How diverse retinal functions arise from feedback at
the first visual synapse. Neuron 99, 117–134.e11 (2018).
TimeShift (s)
MI
(Arbitrary
Unit)
Input stimulus
MI between cone output and input :

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Simulation result of Retina feedback model for different ganglion cell type
RGC output
Simulation
Experiment
Predictive cell
Non-Predictive cell

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Conclusion
1. Cone cell cannot predict without horizontal cell’s feedback
2. Some types of RGC have predictive property
Bipolar
Photoreceptor
Horizontal
cell
𝑥(𝑡)
𝑦(𝑡)
𝑧(𝑡)
Schematic of the model

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Reference
[1] Drinnenberg, A. et al. “How diverse retinal functions arise from feedback at the first visual synapse.” Neuron 99,
117–134.e11 (2018).
[2] Chen, Kevin Sean, Chun-Chung Chen, and C. K. Chan. "Characterization of predictive behavior of a retina by
mutual information." Frontiers in computational neuroscience 11 (2017): 66.
[3] Voss, Henning U. "Signal prediction by anticipatory relaxation dynamics." Physical Review E 93.3 (2016): 030201.

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End
Questions and advices are welcomed

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𝑥𝑡+Δ𝑡 = (1 − Δ𝑡/𝜏)𝑥𝑡+𝜉𝑡 𝐷Δ𝑡
Ornstein-Uhlenbeck process (OU)
This is a Brownian-like walker but this walker cannot move far from the origin. Here 𝜏 is
a time constant which governs the correlation time of the walker.
This OU signal is then low-pass filtered with different cut-off frequencies. The filtered
signals become smoother than the original one.
Appendix 1

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Appendix 2

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R(t): cone response
h(t): horizontal cell feedback
Appendix 3
𝑦 𝑡 = −𝛼𝑦 𝑡 + 𝐾 𝑥 𝑡 − 𝑧 𝑡
𝑧 𝑡 = −𝛽𝑧 𝑡 + 𝑔𝑦 𝑡
𝑦 𝑡 = −𝛼𝑦 𝑡 + 𝐾 𝑥 𝑡 − 𝑦 𝑡 − 𝜏 (Voss, Henning U., 2016)
𝑦 𝑡 − 𝜏 is a delayed negative feedback. It is shown that this model gives rise to prediction because
of the negative group delay (NGD) property the system.
In our case, this delay can come from the inhibition feedback of horizontal cell to the cone cell. To
make the model more structural, the horizontal cell feedback can be describe by a new function 𝑧(𝑡).
The model becomes:
𝛼, 𝛽: relaxation time of 𝑦 𝑡 , 𝑧(𝑡)
𝐾: anticipatory coupling constant
𝑔: input gain of 𝑧(𝑡)