neurons to networks - university of washingtonfaculty.washington.edu/etsb/amath342/materials... ·...
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![Page 1: Neurons to networks - University of Washingtonfaculty.washington.edu/etsb/AMATH342/materials... · max τ τ t t P st Pee − − ... If synapse receives input spikes at rate r, then](https://reader035.vdocuments.site/reader035/viewer/2022070917/5fb7e712d743af1a353e8456/html5/thumbnails/1.jpg)
Neurons to networks
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How do synapses transform inputs?
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Excitatory synapse
Input spike
! Neurotransmitter release
binds to/opens Na channels
Change in synaptic conductance
! Na+ influx
! Depolarization due to EPSP (excitatory postsynaptic potential)
E.g. AMPA synapse
Vocab: Depolarization means make V less neg = more positive
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Input spike
! Neurotransmitter release
binds to/opens K channels
Change in synaptic conductance
K+ leaves cell
! Hyperpolarization due to IPSP (inhibitory postsynaptic potential)
Inhibitory synapse
Vocab: hyperpolarization means make V more negative
![Page 5: Neurons to networks - University of Washingtonfaculty.washington.edu/etsb/AMATH342/materials... · max τ τ t t P st Pee − − ... If synapse receives input spikes at rate r, then](https://reader035.vdocuments.site/reader035/viewer/2022070917/5fb7e712d743af1a353e8456/html5/thumbnails/5.jpg)
Modeling a synaptic input to a (RC) neuron
V
Synaptic conductance
R
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Modeling a synaptic input to a (RC) neuron
V
Synaptic conductance
R
Probability of transmitter release given an input spike
Probability of postsynaptic channel opening(= fraction of channels opened)
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Basic synapse model
Assume Prel = 1 (for now)
What does a single spike input do to Ps?
Kinetic model:
sssss PPdtdP
βα −−= )1(
closed open
open closed
s
s
⎯→⎯
⎯→⎯
β
α
Opening rate Closing rate
Fraction of channels closed
Fraction of channels open
Where:
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Synaptic filters
A difference of exponentials model better fits biological data for GABA,NMDA synapse types
Exponential
Difference of exponentials
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Simplified synaptic models
Difference of exponentials:
Alpha function:
peak
t
peaks ettP τ
τ
−
⋅= const)(
)(const)( 21max
ττtt
s eePtP−−
−⋅=t
Ps
![Page 10: Neurons to networks - University of Washingtonfaculty.washington.edu/etsb/AMATH342/materials... · max τ τ t t P st Pee − − ... If synapse receives input spikes at rate r, then](https://reader035.vdocuments.site/reader035/viewer/2022070917/5fb7e712d743af1a353e8456/html5/thumbnails/10.jpg)
What happens with a sequence of input spikes?
Input
(Markram & Tsodyks, 1998)
Short-term facilitation
Short-term depression
• Biological synapses are dynamic!
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Short-term synaptic plasticity: describe this via P_relRecall definition of synaptic conductance:
Idea: Specify how Prel changes as a
function of consecutive input spikes
srelss PPgg max,=
relrel
P PPdtdP
−= 0τBetween input spikes, Prel decays exponentially back to P0
relDrel PfP → depression: decrement Prel
If input spike:
Abbott et al 1997
Input
[sketch on board]
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Lab exercise:
Write a code that implements the Abbott et al mechanism for synaptic depression.
Drive the synapse with spikes occurring regularly at 20 Hz, as in Fig. 1A of Abbott et al ’97. Can you reproduce that figure?
Hint: this should be a few lines of code.
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Key result (a few lines of calculation, see (7) in paper [ESB notes]):
If synapse receives input spikes at rate r, then the steady state value of
P_rel (r) ~ 1/r
Impact of synaptic depression
![Page 15: Neurons to networks - University of Washingtonfaculty.washington.edu/etsb/AMATH342/materials... · max τ τ t t P st Pee − − ... If synapse receives input spikes at rate r, then](https://reader035.vdocuments.site/reader035/viewer/2022070917/5fb7e712d743af1a353e8456/html5/thumbnails/15.jpg)
Consequences of synaptic depression: steady state
depression
Input Rate
average release probability
average transmission rate
At steady state,
Constant synaptic input for Wide range of inputs!
Steady-state gain control
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Consequences of synaptic depression: dynamic response
Steady-state transmission rates are similar for different rates
Transient inputs are amplified relative to steady-state inputs
Input rates r
In fact: an equal-percent change from baseline gives an equal transient response.Who cares? Abbott et al 97: Neuron gets inputs from 1000’s of upstream cells, each of which fires at 1-200 Hz. How can we be responsive to all?
Synaptic depression yields “gain control” to satisfy this. vs. Adaptation or inhibition, does so in an INPUT-SPECIFIC way!
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Extending the model to include facilitationRecall definition of synaptic conductance:
srelss PPgg max,=
Between input spikes, Prel still decays exponentially back to P0
relDrel PfP →
)1( relFrelrel PfPP −+→
depression: decrement Prel
facilitation: increment Prel
If input spike:
Abbott et al 1997
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Effects of synaptic facilitation & depression
facilitation depression
Input Rate Input Rateaverage release probability
average transmission rate
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More detailed plasticity model
ER
I
• Incoming spike activates a fraction of recovered resources R to become effective (E)
• Amount of effective resources E govern size of gs(t)• Effective resources rapidly inactivate (msec)• Inactivated recover (100s of msec)• E(t) determines postsynaptic current
Markram and Tsodyks, PNAS 1997
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Match between model and experiment
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Synapses provide an additional layer of dynamics and computation, beyond that occurring in single neurons!
Summary:
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Network computation via simplified “firing rate” models
xt ythtStimulus Function
W, connection weight matrix
W_ij = weight from j to i neuron (or neural population) ifires with rate
inputi = stimi(t) +X
j
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neuron (or neural population) ireceives input
ri(t)<latexit sha1_base64="TDxq+riMmEuvG1mwEN4UMd0gCh8=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSLUS0lFUG9FLx4rmLbQhrLZbtq1m03YnQgl9D948aDi1R/kzX/jts1BWx8MPN6bYWZekEhh0HW/nZXVtfWNzcJWcXtnd2+/dHDYNHGqGfdYLGPdDqjhUijuoUDJ24nmNAokbwWj26nfeuLaiFg94DjhfkQHSoSCUbRSU/dEBc96pbJbdWcgy6SWkzLkaPRKX91+zNKIK2SSGtOpuQn6GdUomOSTYjc1PKFsRAe8Y6miETd+Nrt2Qk6t0idhrG0pJDP190RGI2PGUWA7I4pDs+hNxf+8TorhlZ8JlaTIFZsvClNJMCbT10lfaM5Qji2hTAt7K2FDqilDG1DRhlBbfHmZeOfV66p7f1Gu3+RpFOAYTqACNbiEOtxBAzxg8AjP8ApvTuy8OO/Ox7x1xclnjuAPnM8fYceOiA==</latexit><latexit sha1_base64="TDxq+riMmEuvG1mwEN4UMd0gCh8=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSLUS0lFUG9FLx4rmLbQhrLZbtq1m03YnQgl9D948aDi1R/kzX/jts1BWx8MPN6bYWZekEhh0HW/nZXVtfWNzcJWcXtnd2+/dHDYNHGqGfdYLGPdDqjhUijuoUDJ24nmNAokbwWj26nfeuLaiFg94DjhfkQHSoSCUbRSU/dEBc96pbJbdWcgy6SWkzLkaPRKX91+zNKIK2SSGtOpuQn6GdUomOSTYjc1PKFsRAe8Y6miETd+Nrt2Qk6t0idhrG0pJDP190RGI2PGUWA7I4pDs+hNxf+8TorhlZ8JlaTIFZsvClNJMCbT10lfaM5Qji2hTAt7K2FDqilDG1DRhlBbfHmZeOfV66p7f1Gu3+RpFOAYTqACNbiEOtxBAzxg8AjP8ApvTuy8OO/Ox7x1xclnjuAPnM8fYceOiA==</latexit><latexit sha1_base64="TDxq+riMmEuvG1mwEN4UMd0gCh8=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSLUS0lFUG9FLx4rmLbQhrLZbtq1m03YnQgl9D948aDi1R/kzX/jts1BWx8MPN6bYWZekEhh0HW/nZXVtfWNzcJWcXtnd2+/dHDYNHGqGfdYLGPdDqjhUijuoUDJ24nmNAokbwWj26nfeuLaiFg94DjhfkQHSoSCUbRSU/dEBc96pbJbdWcgy6SWkzLkaPRKX91+zNKIK2SSGtOpuQn6GdUomOSTYjc1PKFsRAe8Y6miETd+Nrt2Qk6t0idhrG0pJDP190RGI2PGUWA7I4pDs+hNxf+8TorhlZ8JlaTIFZsvClNJMCbT10lfaM5Qji2hTAt7K2FDqilDG1DRhlBbfHmZeOfV66p7f1Gu3+RpFOAYTqACNbiEOtxBAzxg8AjP8ApvTuy8OO/Ox7x1xclnjuAPnM8fYceOiA==</latexit>
⌧dridt
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DYNAMICS: rates approach steady states f(input)f(input)
input