Transient oscillations in excitable cells
and Spike-adding Mechanism in
Transient Bursts
2016 NZMRI Summer School Lecture 2
Krasimira Tsaneva-Atanasova
University of Exeter
Mechanisms of intrinsic plasticity in the
hippocampus
Methods
• CA3 pyramidal neurons
• P14 wistar rats
• 32-34 ºC
• Kgluconate-based internal solution
• Whole-cell current clamp
recordings
• Resting membrane potential held
constant by automatic slow somatic
current injection
• Experiments performed in blockers
of fast excitatory (AMPA/kainate &
NMDA receptors) and inhibitory
transmission (GABAA receptors)
CA1
20 mV
CA3
DG
200 ms
200 pA
-100 pA
Current injection responses of 36 CA3 pyramidal
neurones at Vrest
100 mV
1s
CA3 pyramids exhibit more profound bursts and larger ADP
+200 pA & -100 pA
500 ms
2 nA
2 ms
CA3
ADP
CA1
The model: Slow-Fast dynamical system
J. Nowacki, H.M. Osinga, J.T. Brown, A.D. Randall, and K.T. Tsaneva-Atanasova, A unified model of CA1/3 pyramidal
cells: An investigation into excitability, Progr. Biophysics and Molecular Biology, 105(1-2), 34-48 (2011)
A unified pyramidal cell model
CA3 vs CA1 neurons
J. Nowacki, H.M. Osinga, J.T. Brown, A.D. Randall, and K.T. Tsaneva-Atanasova, A unified model of CA1/3 pyramidal
cells: An investigation into excitability, Progr. Biophysics and Molecular Biology, 105(1-2), 34-48 (2011)
Increased burstyness and ADP
in aged PSAPP CA1 neurones
1. Higher burst frequency
2. Larger ADP
WT
PSAPP
PSAPP
WT
10 mV
10 ms
*
20
10
0
*
*
100 200 300
Current step (pA)
Instantaneous frequency (Hz)
number of spikes
200
200
100 pA
P<0.001
150
150
100
100
50
50
0
0
2
4
6
8
20
200 pA
P<0.01
10
0
Spike pair #
0
2
4
6
8
ADP amplitude (mV)
WT
PSAPP
30
10
15
*
10
5
0
WT
PSAPP
J.T. Brown, J. Chin, S.C. Leiser, M.N. Pangalos, and A.D. Randall. Altered intrinsic neuronal excitability and reduced Na+
currents in a mouse model of Alzheimer's disease. Neurobiology of Aging, 32(11), 2109-2114 (2011)
Short stimulus responses in experiments
V (mV)
4. Enhanced ADP
40
20
0
-20
-40
-60
-80
Control
50 % GNa
2 mV
20 ms
20 ms
20
I 0
2
(A/cm )
3. Enhanced burstiness
Inst. frequency (Hz)
150
1 μA.cm-2
150
100
100
50
50
0
2 μA.cm-2
0
1 2 3 4 5 6 7
0 2 4 6 8 10
Spike pair #
J.T. Brown, J. Chin, S.C. Leiser, M.N. Pangalos, and A.D. Randall. Altered intrinsic neuronal excitability and reduced Na+
currents in a mouse model of Alzheimer's disease. Neurobiology of Aging, 32(11), 2109-2114 (2011)
The ADP drives high frequency burst firing
400
Action potential
40
60
0
0
0
20 mV
10 ms
Vm (mV)
ADP
Time
20
-60
80
0
-20
-40
Vmem
40
(ms)
V)
60
200
Vm (m
dV/dt (V/s)
2 nA, 2 ms
AP threshold
-60
Imem
-80
2 nA step
2 ms
20 ms
Brown JT, Randall AD. Activity-dependent depression of the spike after-depolarization generates long-lasting intrinsic
plasticity in hippocampal CA3 pyramidal neurons. J Physiol. 587: 1265-81 (2009)
A unified pyramidal cell model
Short stimulus responses of CA3 and CA1 neurons
J. Nowacki, H.M. Osinga, J.T. Brown, A.D. Randall, and K.T. Tsaneva-Atanasova, A unified model of CA1/3 pyramidal
cells: An investigation into excitability, Progr. Biophysics and Molecular Biology, 105(1-2), 34-48 (2011)
Definition of the ADP
400
200
0
0
Time
40
(ms)
-60
80
V)
60
0
Vm (m
dV/dt (V/s)
Local minimum and local maximum
Definition of the ADP
using the first and second derivative of the membrane potential
So far…
 We could define the transient phenomenon of afterdepolarization using the derivatives of the membrane
potential;
 We show that the nullclines of the membrane potential play
a role of a bursting-threshold in the model;
 Transitions between no ADP, ADP and a burst can be studied
in a parameter-dependent setting.
Excitability threshold as a two-point boundary
value problem (reduced model)
(2.1)
Short stimulus responses in the model
J. Nowacki, H.M. Osinga, and K.T. Tsaneva-Atanasova, Continuation-based numerical detection of afterdepolarisation and spike-adding threshold, Neural Computation, 25(4):877-900, (2013)
Excitability threshold as a two-point boundary
value problem (2PBVP)
Action potential
20 mV
ADP
10 ms
Vmem
Imem
2 nA step
2 ms
The definition of the after-depolarization allows us to systematically investigate its onset
and the generation of a burst
Krauskopf B, Osinga HM: Computing invariant manifolds via the continuation of orbit segments. In Numerical
Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems. Edited by Krauskopf
B, Osinga HM, Galán-Vioque J. Dordrecht: Springer; 2007:117-154.
Excitability threshold as a 2PBVP
J. Nowacki, H.M. Osinga, and K.T. Tsaneva-Atanasova, Continuation-based numerical detection of after-depolarisation and
spike-adding threshold, Neural Computation, 25(4):877-900, (2013)
Identifying the onset of ADP in a transient burst
P
B
Identifying spike adding mechanism
J. Nowacki, H.M. Osinga, and K.T. Tsaneva-Atanasova, Dynamical systems analysis of spike-adding mechanisms in
transient bursts, J. Mathematical Neuroscience 2, 7 (2012)
Identifying spike adding mechanism
Onset of the transient busts
ò( u
1
(uON ,uOFF ) 2 =
ON
0
2
(t) + uOFF (t)
2
) dt
J. Nowacki, H.M. Osinga, and K.T. Tsaneva-Atanasova, Dynamical systems analysis of spike-adding mechanisms in
transient bursts, J. Mathematical Neuroscience 2, 7 (2012)
Can we observe transient spike adding in a
three dimensional system?
Transient busts in 3D
dx
= sax 3 - sx 2 - hy - bz + I app
dt
dy
= j (x 2 - y)
dt
dz
= e (sa1x + b1 - kz)
dt
Osinga, Hinke M. and Tsaneva-Atanasova, Krasimira T., 2013, Geometric analysis of transient bursts, Chaos: An
Interdisciplinary Journal of Nonlinear Science, 23, 046107
Onset of the ADP and transient busts in 3D
Osinga, Hinke M. and Tsaneva-Atanasova, Krasimira T., 2013, Geometric analysis of transient bursts, Chaos: An
Interdisciplinary Journal of Nonlinear Science, 23, 046107
Spike onset along saddle-type slow manifold
Spike onset along saddle-type slow manifold
Two-parameter curves of spike-adding onsets
Osinga, Hinke M. and Tsaneva-Atanasova, Krasimira T., 2013, Geometric analysis of transient bursts, Chaos: An
Interdisciplinary Journal of Nonlinear Science, 23, 046107
Onset of the ADP and transient busts
J. Nowacki, H.M. Osinga, and K.T. Tsaneva-Atanasova, Continuation-based numerical detection of afterdepolarisation and spike-adding threshold, Neural Computation, 25(4):877-900, (2013)