Neurons to networks

How do synapses transform inputs?

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

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

Modeling a synaptic input to a (RC) neuron

V

Synaptic conductance

R

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)

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:

Synaptic filters

A difference of exponentials model better fits biological data for GABA,NMDA synapse types

Exponential

Difference of exponentials

Simplified synaptic models

Difference of exponentials:

Alpha function:

peak

t

peaks ettP τ

τ

−

⋅= const)(

)(const)( 21max

ττtt

s eePtP−−

−⋅=t

Ps

What happens with a sequence of input spikes?

Input

(Markram & Tsodyks, 1998)

Short-term facilitation

Short-term depression

• Biological synapses are dynamic!

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]

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.

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

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

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!

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

Effects of synaptic facilitation & depression

facilitation depression

Input Rate Input Rateaverage release probability

average transmission rate

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

Match between model and experiment

Synapses provide an additional layer of dynamics and computation, beyond that occurring in single neurons!

Summary:

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

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⌧dridt

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DYNAMICS: rates approach steady states f(input)f(input)

input