16. T = 1 s
Input duration T (s)
Adaptation
timescale(s)
Input
Firing rate
(Lundstrom et al. Nature Neuroscience 2008)
What is the timescale of adaptation?
17. T = 1 s
~ 0.1 s
Input duration T (s)
Adaptation
timescale(s)
x
(Lundstrom et al. Nature Neuroscience 2008)
Input
Firing rate
What is the timescale of adaptation?
18. T = 10 s
Input duration T (s)
Adaptation
timescale(s)
(Lundstrom et al. Nature Neuroscience 2008)
x
Input
Firing rate
What is the timescale of adaptation?
19. T = 10 s
Input duration T (s)
Adaptation
timescale(s)
x
x
~ 1 s
(Lundstrom et al. Nature Neuroscience 2008)
Input
Firing rate
What is the timescale of adaptation?
20. (Lundstrom et al. Nature Neuroscience 2008)
T = 30 s
Input duration T (s)
Adaptation
timescale(s)
x
x
Input
Firing rate
What is the timescale of adaptation?
21. (Lundstrom et al. Nature Neuroscience 2008)
~ 3 s
T = 30 s
Input duration T (s)
Adaptation
timescale(s)
x
x
x
Input
Firing rate
What is the timescale of adaptation?
22. (Lundstrom et al. Nature Neuroscience 2008)
~ 3 s
T = 30 s
Input duration T (s)
Adaptation
timescale(s)
x
x
x
What is the timescale of adaptation?
Input
Firing rate
23. Input
Firing rate
(Lundstrom et al. Nature Neuroscience 2008)
~ 3 s
T = 30 s
Input duration T (s)
Adaptation
timescale(s)
x
x
x
Input duration T (s)
Adaptation
timescale(s)
The timescale of adaptation is not
fixed but depends on the input
What is the timescale of adaptation?
24. Input
Firing rate
(Lundstrom et al. Nature Neuroscience 2008)
~ 3 s
T = 30 s
Input duration T (s)
Adaptation
timescale(s)
x
x
x
Input duration T (s)
Adaptation
timescale(s)
The timescale of adaptation is not
fixed but depends on the input
What is the timescale of adaptation?
Standard
spiking models
25. Period T (s)Period T (s)
Gain(Hz/pA)
Phase(deg)
A different protocol to assess scale-free adaptation
Input
Firing rate
Period T
(Lundstrom et al. Nature Neuroscience 2008)
26. Period T (s)Period T (s)
Gain(Hz/pA)
Phase(deg)
A different protocol to assess scale-free adaptation
Input
Firing rate
Phase response
Period T
(Lundstrom et al. Nature Neuroscience 2008)
27. Period T (s)Period T (s)
Gain(Hz/pA)
Phase(deg)
A different protocol to assess scale-free adaptation
Input
Firing rate
Phase response
Period T
Gain
(Lundstrom et al. Nature Neuroscience 2008)
28. H(w) = w↵
· ei ⇡
2 ↵
= (iw)↵
Period T (s)Period T (s)
Gain(Hz/pA)
Phase(deg)
↵ ⇡ 0.14
↵ ⇡ 0.14
(Lundstrom et al. Nature Neuroscience 2008)
A different protocol to assess scale-free adaptation
29. H(w) = w↵
· ei ⇡
2 ↵
= (iw)↵
Period T (s)Period T (s)
Gain(Hz/pA)
Phase(deg)
Power-law gain
(high-pass filtering)
↵ ⇡ 0.14
↵ ⇡ 0.14
(Lundstrom et al. Nature Neuroscience 2008)
A different protocol to assess scale-free adaptation
30. H(w) = w↵
· ei ⇡
2 ↵
= (iw)↵
Period T (s)Period T (s)
Gain(Hz/pA)
Phase(deg)
Constant phase lead
(scale invariance)
Power-law gain
(high-pass filtering)
↵ ⇡ 0.14
↵ ⇡ 0.14
(Lundstrom et al. Nature Neuroscience 2008)
A different protocol to assess scale-free adaptation
31. A different protocol to assess scale-free adaptation
Power-law gain
(high-pass filtering)
Constant phase lead
(scale invariance)
H(w) = w↵
· ei ⇡
2 ↵
= (iw)↵
Period T (s)Period T (s)
Gain(Hz/pA)
Phase(deg)
↵ ⇡ 0.14
↵ ⇡ 0.14
(Lundstrom et al. Nature Neuroscience 2008)
Pyr NeuronInput current Firing rate
• Linear rate model
• Does not explain the origin of scale invariance
H(w)
Elegant and compact description but...
32. A different protocol to assess scale-free adaptation
Power-law gain
(high-pass filtering)
Constant phase lead
(scale invariance)
H(w) = w↵
· ei ⇡
2 ↵
= (iw)↵
Period T (s)Period T (s)
Gain(Hz/pA)
Phase(deg)
↵ ⇡ 0.14
↵ ⇡ 0.14
(Lundstrom et al. Nature Neuroscience 2008)
Pyr NeuronInput current Spikes
H(w)
We would like a Spiking Neuron Model
that captures scale-free adaptation...
?
33. Outline
PART I: Spike-frequency adaptation
• Introduction: scale-free adaptation
• Spiking neuron model (GIF)
• Functional implications
PART II: Enhanced sensitivity to input fluctuations
• Introduction: f-I curves
• Spiking neuron model (iGIF)
• Functional implications
43. Stochastic spiking mechanism
Spikes are emitted stochastically according to the escape rate mechanism:
(t) = 0 · exp
✓
V VT
V
◆
V
VT
Time
P{ˆt 2 [t; t + T]} ⇡ (t) T
44. Stochastic spiking mechanism
Spikes are emitted stochastically according to the escape rate mechanism:
(t) = 0 · exp
✓
V VT
V
◆
V
VT
Time
P{ˆt 2 [t; t + T]} ⇡ (t) T
Subthreshold dynamics of the membrane potential
Leaky integration plus spike-triggered adaptation current
C
dV
dt
= gL(V EL) + I
X
ˆt<t
⌘(t ˆt)
Time after
spike
⌘
C
dV
dt
= gL(V EL) + I
X
ˆt<t
⌘(t ˆt)
45. Stochastic spiking mechanism
Spikes are emitted stochastically according to the escape rate mechanism:
(t) = 0 · exp
✓
V VT
V
◆
V
VT
Time
P{ˆt 2 [t; t + T]} ⇡ (t) T
Subthreshold dynamics of the membrane potential
Leaky integration plus spike-triggered adaptation current
C
dV
dt
= gL(V EL) + I
X
ˆt<t
⌘(t ˆt)
Time after
spike
⌘
C
dV
dt
= gL(V EL) + I
X
ˆt<t
⌘(t ˆt)
Stochastic spiking mechanism
Spikes are emitted stochastically with conditional firing intensity (escape-rate):
(t) = 0 · exp
✓
V VT
V
◆
V
VT
Time
P{ˆt 2 [t; t + T]} ⇡ (t) T
Dynamics of the firing threshold
Baseline plus spike-triggered movements
Time after
spike
VT (t) = V ⇤
T +
X
ˆt<t
(t ˆt)
46. GIF model parameter extraction
Step 1: Spike-triggered adaptation current
⌘
Time
Adaptation
current
Time
Adaptation
current
Linear expansion in
rectangular basis functions
⌘(t) =
X
i
bi · fi(t)
Step 2: Spike-triggered threshold movement
Time
Moving
threshold
Time
Moving
threshold
Linear expansion in
rectangular basis functions
(t) =
X
i
ci · fi(t)
47. GIF model parameter extraction
Step 1: Spike-triggered adaptation current
⌘
Time
Adaptation
current
Time
Adaptation
current
Linear expansion in
rectangular basis functions
⌘(t) =
X
i
bi · fi(t)
Step 2: Spike-triggered threshold movement
Time
Moving
threshold
Time
Moving
threshold
Linear expansion in
rectangular basis functions
(t) =
X
i
ci · fi(t)
48. GIF model parameter extraction
Step 1: Spike-triggered adaptation current
⌘
Time
Adaptation
current
Time
Adaptation
current
Linear expansion in
rectangular basis functions
⌘(t) =
X
i
bi · fi(t)
Step 2: Spike-triggered threshold movement
Time
Moving
threshold
Time
Moving
threshold
Linear expansion in
rectangular basis functions
(t) =
X
i
ci · fi(t)
49. GIF model parameter extraction
Step 1: Spike-triggered adaptation current
⌘
Time
Adaptation
current
Time
Adaptation
current
Linear expansion in
rectangular basis functions
⌘(t) =
X
i
bi · fi(t)
Step 2: Spike-triggered threshold movement
Time
Moving
threshold
Time
Moving
threshold
Linear expansion in
rectangular basis functions
(t) =
X
i
ci · fi(t)
Least-square linear regression
on the voltage derivative (convex)
50. GIF model parameter extraction
Step 1: Spike-triggered adaptation current
⌘
Time
Adaptation
current
Time
Adaptation
current
Linear expansion in
rectangular basis functions
⌘(t) =
X
i
bi · fi(t)
Step 2: Spike-triggered threshold movement
Time
Moving
threshold
Time
Moving
threshold
Linear expansion in
rectangular basis functions
(t) =
X
i
ci · fi(t)
Least-square linear regression
on the voltage derivative (convex)
Maximum likelihood (convex)
51. Results I
Each spike triggers both an adaptation current and a movement of the
threshold that last for > 20 seconds and decay according to a power-law
52. Results I
⌘(t) / t 0.76
Each spike triggers both an adaptation current and a movement of the
threshold that last for > 20 seconds and decay according to a power-law
53. Results I
⌘(t) / t 0.76
Each spike triggers both an adaptation current and a movement of the
threshold that last for > 20 seconds and decay according to a power-law
54. Results I
(t) / t 0.87
Each spike triggers both an adaptation current and a movement of the
threshold that last for > 20 seconds and decay according to a power-law
⌘(t) / t 0.76
55. Results II
The GIF model with power-law spike-triggered adaptation
captures the experimental transfer function.
Frequency (Hz)
Gain(Hz/nA)
102.1
10
2.0
10
1.9
10
2.2
10 -1.5
10 -1.0
10 -0.5
10 0.0
10 0.5
Data
(Pozzorini et al. Nature Neuroscience 2013)
56. Results II
The GIF model with power-law spike-triggered adaptation
captures the experimental transfer function.
Frequency (Hz)
Gain(Hz/nA)
102.1
10
2.0
10
1.9
10
2.2
10 -1.5
10 -1.0
10 -0.5
10 0.0
10 0.5
Data
GIF
(Pozzorini et al. Nature Neuroscience 2013)
59. The GIF model accurately predicts the occurrence of individual spikes
with ms precision....
(Pozzorini et al., Nature Neuroscience 2013)
Results III
500 ms
60. The GIF model accurately predicts the occurrence of individual spikes
with ms precision....
(Pozzorini et al., Nature Neuroscience 2013)
Results III
500 ms
61. The GIF model accurately predicts the occurrence of individual spikes
with ms precision....
(Pozzorini et al., Nature Neuroscience 2013)
Results III
500 ms
80 % spikes
correctly predicted
(4 ms precision)
0.0
0.5
1.0
M∗
d(−)
GIF
62. Results III
GIF model with 1-s-long spike-triggered adaptation does
not capture the experimental data.
Frequency (Hz)
Gain(Hz/nA)
102.1
10
2.0
10
1.9
10
2.2
10 -1.5
10 -1.0
10 -0.5
10 0.0
10 0.5
Data
GIF
(Pozzorini et al. Nature Neuroscience 2013)
63. Results III
GIF model with 1-s-long spike-triggered adaptation does
not capture the experimental data.
Frequency (Hz)
Gain(Hz/nA)
102.1
10
2.0
10
1.9
10
2.2
10 -1.5
10 -1.0
10 -0.5
10 0.0
10 0.5
Data
GIF
Control
(Pozzorini et al. Nature Neuroscience 2013)
64. Outline
PART I: Spike-frequency adaptation
• Introduction: scale-free adaptation
• Spiking neuron model (GIF)
• Functional implications
PART II: Enhanced sensitivity to input fluctuations
• Introduction: f-I curves
• Spiking neuron model (iGIF)
• Functional implications
67. Functional role
OutputInput
Frequency (Hz)
Flat Power
spectrum
Power
No temporal correlations
Prediction
Barlow’s hypothesis (1961): in the brain, information is represented (encoded)
in an efficient way. Redundant information should be suppressed.
68. Functional role
OutputInput
Frequency (Hz)
Flat Power
spectrum
Power
Frequency (Hz)
High-pass
filtering
Gain
Barlow’s hypothesis (1961): in the brain, information is represented (encoded)
in an efficient way. Redundant information should be suppressed.
69. Functional role
OutputInput
Frequency (Hz)
Flat Power
spectrum
Power
Frequency (Hz)
High-pass
filtering
Gain
Frequency (Hz)
Power-law
spectrum
Power
=x
Barlow’s hypothesis (1961): in the brain, information is represented (encoded)
in an efficient way. Redundant information should be suppressed.
72. Temporal whitening
2
2
Model
Power-law spike-frequency adaptation is optimally tuned to remove
temporal correlations of the input.
GIF model
Voltage (model)
Input current
In vivo data
PI(f) / f 0.67
Scale-free Gaussian process
(power-law temporal correlation)
75. Conclusion I
• Scale-free adaptation is implemented by two power-law
spike-triggered processes that last for > 20 s
• “Natural input” received in vivo by neocortical neurons
fluctuates on multiple timescales (scale-free)
• The multiple timescales of adaptation are optimally tuned
to remove temporal correlations in the input
• Power-law adaptation maximizes the information transfer
76. Outline
PART I: Spike-frequency adaptation
• Introduction: scale-free adaptation
• Spiking neuron model (GIF)
• Functional implications
PART II: Enhanced sensitivity to input fluctuations
• Introduction: f-I curves
• Spiking neuron model (iGIF)
• Functional implications
78. OutputInput
Single-neuron sensitivity to input
fluctuations
Experiment: present a set of stationary input
currents generated by systematically varying
the mean and the variance.
79. OutputInput GIF model
Experiment: present a set stationary input currents
generated by systematically varying the mean and
the variance.
Input st. dev.
0 pA
50 pA
100 pA
150 pA
Single-neuron sensitivity to input
fluctuations
80. OutputInput GIF model
Experiment: present a set stationary input currents
generated by systematically varying the mean and
the variance.
Input st. dev.
0 pA
50 pA
100 pA
150 pA
Subthreshold regime:
sensitivity to input fluctuations
Single-neuron sensitivity to input
fluctuations
81. OutputInput GIF model
Experiment: present a set stationary input currents
generated by systematically varying the mean and
the variance.
Input st. dev.
0 pA
50 pA
100 pA
150 pA
Subthreshold regime:
sensitivity to input fluctuations
Suprathreshold regime:
no sensitivity to input fluctuations
Single-neuron sensitivity to input
fluctuations
82. Single-neuron sensitivity to input
fluctuations
Experimental data shows that single neurons maintain sensitivity to input
fluctuations even in the suprathreshold regime.
1.2
Mean input (nA)
0.6
Firingrate(Hz)
50 pA
150 pA
300 pA
0.0
L5 Pyr Neurons
Rat Prefrontal Cortex
(Arsiero et al., 2007)
83. Single-neuron sensitivity to input
fluctuations
Experimental data shows that single neurons maintain sensitivity to input
fluctuations even in the suprathreshold regime.
1.2
Mean input (nA)
0.6
Firingrate(Hz)
50 pA
150 pA
300 pA
0.0
L5 Pyr Neurons
Rat Prefrontal Cortex
(Arsiero et al., 2007)
CA1 Pyr Neurons
Rat Hippocampus
(Fernandez et al., 2011)
Firingrate(Hz)
Mean input (nA)
84. 0 pA
50 pA
100 pA
150 pA
Results I
We measured the f-I curves of L5 Pyr neurons (mice SSC) by varying
the magnitude of input fluctuations.
85. 0 pA
50 pA
100 pA
150 pA
Enhanced sensitivity
to input fluctuations
Results I
We measured the f-I curves of L5 Pyr neurons (mice SSC) by varying
the magnitude of input fluctuations.
86. Results I
0 pA
50 pA
100 pA
150 pA
We measured the f-I curves of L5 Pyr neurons (mice SSC) by varying
the magnitude of input fluctuations
Voltage threshold
for spike initiation
87. Results I
0 pA
50 pA
100 pA
150 pA
We measured the f-I curves of L5 Pyr neurons (mice SSC) by varying
the magnitude of input fluctuations
Consistent with spike-triggered
threshold movement
88. Results I
0 pA
50 pA
100 pA
150 pA
We measured the f-I curves of L5 Pyr neurons (mice SSC) by varying
the magnitude of input fluctuations
89. Results I
0 pA
50 pA
100 pA
150 pA
We measured the f-I curves of L5 Pyr neurons (mice SSC) by varying
the magnitude of input fluctuations
Firing threshold is reduced in
presence of input fluctuations
90. Results I
0 pA
50 pA
100 pA
150 pA
We measured the f-I curves of L5 Pyr neurons (mice SSC) by varying
the magnitude of input fluctuations
Firing threshold is reduced in
presence of input fluctuations
The firing threshold dynamics of the
GIF model has to be extended!
VT (t) = V ⇤
T +
X
ˆt<t
(t ˆt) + . . .
91. (Azouz and Gray, Neuron 2003)
Experimental evidence
The voltage threshold for spike initiation depends on the speed at which
the threshold is reached.
92. Theoretical prediction
Fast Na-channel inactivation implements a nonlinear coupling between
membrane potential and firing threshold.
(Platkiewicz and Brette, PLOS Comp. 2011)
✓1(V )⌧✓
˙✓ = ✓ +
✓1(V )
Firingthreshold(mV)✓
Membrane potential (mV)V
93. spike
V > ✓
Theoretical prediction
Fast Na-channel inactivation implements a nonlinear coupling between
membrane potential and firing threshold.
(Platkiewicz and Brette, PLOS Comp. 2011)
✓1(V )⌧✓
˙✓ = ✓ +
✓1(V )
Firingthreshold(mV)✓
Membrane potential (mV)V
94. spike
V > ✓
Theoretical prediction
Fast Na-channel inactivation implements a nonlinear coupling between
membrane potential and firing threshold.
(Platkiewicz and Brette, PLOS Comp. 2011)
✓1(V )⌧✓
˙✓ = ✓ +
Responses to different
depolarization ramps
dV
dt
✓1(V )
Firingthreshold(mV)✓
Membrane potential (mV)V
95. spike
V > ✓
Theoretical prediction
Fast Na-channel inactivation implements a nonlinear coupling between
membrane potential and firing threshold.
(Platkiewicz and Brette, PLOS Comp. 2011)
✓1(V )⌧✓
˙✓ = ✓ +
dV
dt
✓1(V )
Firingthreshold(mV)✓
Membrane potential (mV)V
96. spike
V > ✓
Theoretical prediction
Fast Na-channel inactivation implements a nonlinear coupling between
membrane potential and firing threshold.
(Platkiewicz and Brette, PLOS Comp. 2011)
✓1(V )⌧✓
˙✓ = ✓ +
dV
dt
✓1(V )
Firingthreshold(mV)✓
Membrane potential (mV)V
97. spike
V > ✓
Theoretical prediction
Fast Na-channel inactivation implements a nonlinear coupling between
membrane potential and firing threshold.
(Platkiewicz and Brette, PLOS Comp. 2011)
✓1(V )⌧✓
˙✓ = ✓ +
dV
dt
✓1(V )
Firingthreshold(mV)✓
Membrane potential (mV)V
98. spike
V > ✓
Theoretical prediction
Fast Na-channel inactivation implements a nonlinear coupling between
membrane potential and firing threshold.
(Platkiewicz and Brette, PLOS Comp. 2011)
✓1(V )⌧✓
˙✓ = ✓ +
dV
dt
✓1(V )
Firingthreshold(mV)✓
Membrane potential (mV)V
99. spike
V > ✓
Theoretical prediction
Fast Na-channel inactivation implements a nonlinear coupling between
membrane potential and firing threshold.
(Platkiewicz and Brette, PLOS Comp. 2011)
✓1(V )⌧✓
˙✓ = ✓ +
dV
dt
✓1(V )
Firingthreshold(mV)✓
Membrane potential (mV)V
100. Outline
PART I: Spike-frequency adaptation
• Introduction: scale-free adaptation
• Spiking neuron model (GIF)
• Functional implications
PART II: Enhanced sensitivity to input fluctuations
• Introduction: f-I curves
• Spiking neuron model (iGIF)
• Functional implications
101. Inactivating Generalized Integrate & Fire model (iGIF)
VT (t) = V ⇤
T +
X
ˆt<t
(t ˆt)
The GIF model is extended by including a nonlinear coupling between membrane
potential and firing threshold:
VT
Vm
Vm
Spikes
Input
Current
Spike-triggered
current
m
Membrane
filter
Spike-triggered
threshold movement
Threshold
filter
Threshold
coupling
Escape-rate
nonlinearity
Spiking
mechanism
∞
Iext
102. Inactivating Generalized Integrate & Fire model (iGIF)
VT (t) = V ⇤
T +
X
ˆt<t
(t ˆt)
The GIF model is extended by including a nonlinear coupling between membrane
potential and firing threshold:
VT
Vm
Vm
Spikes
Input
Current
Spike-triggered
current
m
Membrane
filter
Spike-triggered
threshold movement
Threshold
filter
Threshold
coupling
Escape-rate
nonlinearity
Spiking
mechanism
∞
Iext
⌧✓
˙✓ = ✓ + ✓1(V )
+ ✓(t)
103. Inactivating Generalized Integrate & Fire model (iGIF)
VT (t) = V ⇤
T +
X
ˆt<t
(t ˆt)
The GIF model is extended by including a nonlinear coupling between membrane
potential and firing threshold:
VT
Vm
Vm
Spikes
Input
Current
Spike-triggered
current
m
Membrane
filter
Spike-triggered
threshold movement
Threshold
filter
Threshold
coupling
Escape-rate
nonlinearity
Spiking
mechanism
∞
Iext
⌧✓
˙✓ = ✓ + ✓1(V )
+ ✓(t)
Extracted from data with a new
nonparametric max-likelihood procedure
V
✓1(V)
104. Evidence for a nonlinear coupling between
membrane potential and firing threshold
Thresholdmovement(mV)Spike-triggeredcurrent(nA)
Power-law
spike-triggered
adaptation
105. Evidence for a nonlinear coupling between
membrane potential and firing threshold
Thresholdmovement(mV)Spike-triggeredcurrent(nA)
Nonlinearthresholdcoupling(mV)
Coupling
timescale(ms)
⌧✓
107. The iGIF model captures enhanced sensitivity to input
fluctuations and improve spike-timing prediction
0 pA
50 pA
100 pA
150 pA
Input fluctuations
108. The iGIF model captures enhanced sensitivity to input
fluctuations and improve spike-timing prediction
0 pA
50 pA
100 pA
150 pA
Input fluctuations
109. The iGIF model captures enhanced sensitivity to input
fluctuations and improve spike-timing prediction
0 pA
50 pA
100 pA
150 pA
Input fluctuations
110. 1 second
GIF iGIF
M⇤
d()
0.0
0.5
1.0
The iGIF model captures enhanced sensitivity to input
fluctuations and improve spike-timing prediction
0 pA
50 pA
100 pA
150 pA
Input fluctuations
111. 1 second
GIF iGIF
M⇤
d()
0.0
0.5
1.0
The iGIF model captures enhanced sensitivity to input
fluctuations and improve spike-timing prediction
0 pA
50 pA
100 pA
150 pA
Input fluctuations
112. 1 second
GIF iGIF
M⇤
d()
0.0
0.5
1.0
The iGIF model captures enhanced sensitivity to input
fluctuations and improve spike-timing prediction
***
86 % spikes correctly
predicted (4 ms precision)
0 pA
50 pA
100 pA
150 pA
Input fluctuations
113. In Pyramidal neurons, the firing threshold dynamics
depends on:
1. Spike-history (multiple timescales)
2. Membrane potential (fast coupling)
Summary
114. In Pyramidal neurons, the firing threshold dynamics
depends on:
1. Spike-history (multiple timescales)
2. Membrane potential (fast coupling)
Summary
Nontrivial interaction
115. V ⇤
T (t) = ✓(t) +
X
ˆt<t
(t ˆt)
Nonlinearthresholdcoupling(mV)
V
=
✓
Nonlinear interaction between spike-dependent and
voltage-dependent threshold changes
116. V ⇤
T (t) = ✓(t) +
X
ˆt<t
(t ˆt)
Nonlinearthresholdcoupling(mV)
V
=
✓
Nonlinear interaction between spike-dependent and
voltage-dependent threshold changes
117. V ⇤
T (t) = ✓(t) +
X
ˆt<t
(t ˆt)
Nonlinearthresholdcoupling(mV)
V
=
✓
Nonlinear interaction between spike-dependent and
voltage-dependent threshold changes
In the absence of spikes
the threshold-voltage
coupling is not active
Masked
118. V ⇤
T (t) = ✓(t) +
X
ˆt<t
(t ˆt)
Nonlinearthresholdcoupling(mV)
V
=
✓
Nonlinear interaction between spike-dependent and
voltage-dependent threshold changes
119. V ⇤
T (t) = ✓(t) +
X
ˆt<t
(t ˆt)
Nonlinearthresholdcoupling(mV)
V
=
✓
+X
ˆt<
t(t
ˆt)
Nonlinear interaction between spike-dependent and
voltage-dependent threshold changes
Spike-triggered threshold
adaptation activates the
threshold-voltage coupling
Unmasked
120. Nonlinear interaction between spike-dependent and
voltage-dependent threshold changes
200 ms
Weak input
200 ms
Medium input
200 ms
Strong input
121. Nonlinear interaction between spike-dependent and
voltage-dependent threshold changes
200 ms
Weak input
200 ms
Medium input
200 ms
Strong input
Thresholdcoupling(mV)
Membrane Potential (mV)
122. Nonlinear interaction between spike-dependent and
voltage-dependent threshold changes
200 ms
Weak input
200 ms
Medium input
200 ms
Strong input
Thresholdcoupling(mV)
Membrane Potential (mV) Membrane Potential (mV)
Thresholdcoupling(mV)
123. Nonlinear interaction between spike-dependent and
voltage-dependent threshold changes
200 ms
Weak input
200 ms
Medium input
200 ms
Strong input
Thresholdcoupling(mV)
Membrane Potential (mV) Membrane Potential (mV)
Thresholdcoupling(mV)
Membrane Potential (mV)
Thresholdcoupling(mV)
124. Nonlinear interaction between spike-dependent and
voltage-dependent threshold changes
200 ms
Weak input
200 ms
Medium input
200 ms
Strong input
Masked Unmasked
Thresholdcoupling(mV)
Membrane Potential (mV) Membrane Potential (mV)
Thresholdcoupling(mV)
Membrane Potential (mV)
Thresholdcoupling(mV)
125. Outline
PART I: Spike-frequency adaptation
• Introduction: scale-free adaptation
• Spiking neuron model (GIF)
• Functional implications
PART II: Enhanced sensitivity to input fluctuations
• Introduction: f-I curves
• Spiking neuron model (iGIF)
• Functional implications
131. To understand the functional role of the nonlinear coupling between
membrane potential and firing threshold it is convenient to reduce the
iGIF model to a GLM
Functional implications
e↵(t) = m(t) · ✓(t) ⇤ m(t)
132. To understand the functional role of the nonlinear coupling between
membrane potential and firing threshold it is convenient to reduce the
iGIF model to a GLM
Functional implications
Average strength of
the thresold-voltage coupling
e↵(t) = m(t) · ✓(t) ⇤ m(t)
133. To understand the functional role of the nonlinear coupling between
membrane potential and firing threshold it is convenient to reduce the
iGIF model to a GLM
Functional implications
e↵(t) = m(t) · ✓(t) ⇤ m(t)
134. To understand the functional role of the nonlinear coupling between
membrane potential and firing threshold it is convenient to reduce the
iGIF model to a GLM
200 ms
Weak input
200 ms
Medium input
200 ms
Strong input
Functional implications
e↵(t) = m(t) · ✓(t) ⇤ m(t)
135. To understand the functional role of the nonlinear coupling between
membrane potential and firing threshold it is convenient to reduce the
iGIF model to a GLM
200 ms
Weak input
200 ms
Medium input
200 ms
Strong input
e↵(t)
0 75 150
2 M /ms
Time (ms)
⇡ 0.0
Functional implications
e↵(t) = m(t) · ✓(t) ⇤ m(t)
136. To understand the functional role of the nonlinear coupling between
membrane potential and firing threshold it is convenient to reduce the
iGIF model to a GLM
200 ms
Weak input
200 ms
Medium input
200 ms
Strong input
e↵(t)
0 75 150
2 M /ms
Time (ms)
⇡ 0.0
0 75 150
2 M /ms
Time (ms)
⇡ 0.5
Functional implications
e↵(t) = m(t) · ✓(t) ⇤ m(t)
137. To understand the functional role of the nonlinear coupling between
membrane potential and firing threshold it is convenient to reduce the
iGIF model to a GLM
200 ms
Weak input
200 ms
Medium input
200 ms
Strong input
e↵(t)
0 75 150
2 M /ms
Time (ms)
⇡ 0.0
0 75 150
2 M /ms
Time (ms)
⇡ 0.5
0 75 150
2 M /ms
Time (ms)
⇡ 1.0
Functional implications
e↵(t) = m(t) · ✓(t) ⇤ m(t)
138. To understand the functional role of the nonlinear coupling between
membrane potential and firing threshold it is convenient to reduce the
iGIF model to a GLM
200 ms
Weak input
200 ms
Medium input
200 ms
Strong input
Integration Differentiation
e↵(t)
0 75 150
2 M /ms
Time (ms)
⇡ 0.0
0 75 150
2 M /ms
Time (ms)
⇡ 0.5
0 75 150
2 M /ms
Time (ms)
⇡ 1.0
Functional implications
e↵(t) = m(t) · ✓(t) ⇤ m(t)
140. iGIF model prediction
Effectivelinearfilterseff(M/ms)
Time (ms)
The shape of the effective filter adaptively changes
depending on the input strength
Experimental f-I curves
Mean input (nA)
0.0 0.60.3
(t) ⇡ 0 exp
0
@C + e↵ ⇤ I(t)
X
ˆtj
he↵(t ˆtj)
1
A
141. Effectivelinearfilterseff(M/ms)
Time (ms)
The shape of the effective filter adaptively changes
depending on the input strength
Mean input (nA)
0.0 0.60.3
iGIF model prediction
(t) ⇡ 0 exp
0
@C + e↵ ⇤ I(t)
X
ˆtj
he↵(t ˆtj)
1
A
142. Effectivelinearfilterseff(M/ms)
Time (ms)
Experimental data
The shape of the effective filter adaptively changes
depending on the input strength
Time (ms)
Filtersextractedfromdata(a.u.)
Weak input
Medium input
Strong input
Mean input (nA)
0.0 0.60.3
iGIF model prediction
(t) ⇡ 0 exp
0
@C + e↵ ⇤ I(t)
X
ˆtj
he↵(t ˆtj)
1
A
143. Effectivelinearfilterseff(M/ms)
Time (ms)
Experimental data
The shape of the effective filter adaptively changes
depending on the input strength
Time (ms)
Filtersextractedfromdata(a.u.)
Weak input
Medium input
Strong input
Mean input (nA)
0.0 0.60.3
iGIF model prediction
(t) ⇡ 0 exp
0
@C + e↵ ⇤ I(t)
X
ˆtj
he↵(t ˆtj)
1
A
144. Effectivelinearfilterseff(M/ms)
Time (ms)
Experimental data
The shape of the effective filter adaptively changes
depending on the input strength
Time (ms)
Filtersextractedfromdata(a.u.)
Weak input
Medium input
Strong input
Mean input (nA)
0.0 0.60.3
iGIF model prediction
(t) ⇡ 0 exp
0
@C + e↵ ⇤ I(t)
X
ˆtj
he↵(t ˆtj)
1
A
145. Somatic integration depends on the input statistics
Generalized Linear Model (GLM)
GLM
SpikesInput eff
Effective
linear filter
Exponential
nonlinearity
Spike-history
filter
hGLM
Iext
Spiking
mechanism
(t) ⇡ 0 exp
0
@C + e↵ ⇤ I(t)
X
ˆtj
he↵(t ˆtj)
1
A
146. Somatic integration depends on the input statistics
Generalized Linear Model (GLM)
GLM
SpikesInput eff
Effective
linear filter
Exponential
nonlinearity
Spike-history
filter
hGLM
Iext
Spiking
mechanism
(t) ⇡ 0 exp
0
@C + e↵ ⇤ I(t)
X
ˆtj
he↵(t ˆtj)
1
A
147. Somatic integration depends on the input statistics
Generalized Linear Model (GLM)
GLM
SpikesInput eff
Effective
linear filter
Exponential
nonlinearity
Spike-history
filter
hGLM
Iext
Spiking
mechanism
(t) ⇡ 0 exp
0
@C + e↵ ⇤ I(t)
X
ˆtj
he↵(t ˆtj)
1
A
148. Somatic integration depends on the input statistics
0.0 0.450.15 0.3
0.1
0.0
0.3
0.2
Mean input (nA)
Inputfluctuations(nA)
(t) ⇡ 0 exp
0
@C + e↵ ⇤ I(t)
X
ˆtj
he↵(t ˆtj)
1
A
149. Conclusion II
• The firing threshold dynamics depends on both previous
spikes and subthreshold voltage
• The interaction between these two mechanisms explains
enhanced sensitivity to input fluctuations
• Depending on the input statistics, somatic integration
switches between leaky integration and coincidence
detection (or differentiation)
