Rhythms
Christoph Cortical Brgers o
Mathematics Department Tufts University Medford, Massachusetts
Many thanks to Bard and Jon for inviting me, and to all of you for listening!
Classication of rhythms by frequency
Gamma (30 90 Hz) Beta (12 30 Hz) Alpha (7 12 Hz) Theta (3 7 Hz) Delta (0.5 3 Hz) (deep sleep) Very fast oscillations (90 200 Hz) (hippocampus and neocortex) (These ranges are agreed upon only approximately.)
Gamma rhythms (3090 Hz)
An example of gamma oscillations in the EEG
O. Bertrand, C. Tallon-Baudry, C. Fischer, and J. Pernier, Object representation and gamma oscillations, Proc. 12th International Conference on Biomagnetism 2001
50-ms, 1000-Hz tone bursts are presented to subjects watching a silent movie. An evoked gamma frequency oscillation is seen in the EEG within about 100 ms, followed by an induced gamma response after about 300 ms.
An example of gamma oscillations in local eld potentials
P. Fries, J. Reynolds, A. Rorie, and R. Desimone, Modulation of oscillatory neuronal synchronization by selective visual attention, Science 2001.
Local eld potentials from area V4 (extrastriate visual cortex) of a monkey:
An example of gamma oscillations in an intracellular recording in a hippocampal slice
M. Whittington, R. Traub, and J. Jeerys, Synchronized oscillations in interneuron networks driven by metabotrobic glutamate receptor activation, Nature 1995
In a hippocampal slice, metabotropic glutamate receptors are activated. This drives fast-spiking interneurons. The result is a gamma oscillation, seen here in an intracellular recording from a pyramidal cell in CA1:
Interneuronal network gamma (ING)
This is the model proposed in
M. Whittington, R. Traub, and J. Jeerys, Synchronized oscillations in interneuron networks driven by metabotrobic glutamate receptor activation, Nature 1995.
According to the ING model, gamma oscillations are generated in networks of fast-spiking inhibitory interneurons with GABAA -receptor mediated mutual synaptic inhibition. Excitatory cells play no role in generating the rhythm.
(picture from Bartos, Vida, and Jonas, Nature Reviews Neuroscience 2007)
Two experimental facts supporting the ING model
The gamma oscillations induced by activation of metabotropic glutamate receptors in CA1 slices are abolished by bicuculline, which blocks GABAA receptors, but maintaind in the presence of drugs blocking AMPA receptors.
Pure ING lacks robustness
X.-J. Wang and G. Buzski, Gamma oscillation by synaptic inhibition in a a hippocampal interneuronal network model, Journal of Neuroscience 1996 J. White et al., Synchronization and oscillatory dynamics in heterogeneous, mutually inhibited neurons, Journal of Computational Neuroscience 1998
When the I-cells (fast-spiking inhibitory interneurons) or the network connections are just slightly heterogeneous, ING rhythms break down. For illustration, here is a numerically simulated ING rhythm with all-to-all connectivity among I-cells:
I!cells 20
0 0
50
100 time [msec]
150
200
... and here is the same with slight randomness in the connectivity:
I!cells 20
0 0
50
100 time [msec]
150
200
Dendritic gap junctions stabilize ING
Traub et al., Gap junctions between interneuron dendrites can enhance synchrony of gamma oscillations in distributed networks, Journal of Neuroscience 2001 N. Kopell and B. Ermentrout, Chemical and electrical synapses perform complementary roles in the synchronization of interneuronal networks, PNAS 2004
With weak gap junctions,
I!cells 20
0 0
50
100 time [msec]
150
200
reverts to
I!cells 20
0 0
50
100 time [msec]
150
200
A second way of stabilizing the rhythm: Adding pyramidal cells, thereby turning ING into PING
R. Traub, M. Whittington, I. Stanford, and J. Jeerys, A mechanism for generation of long-range synchronous fast oscillations in the cortex, Nature 1996 M. Whittington et al., Spatiotemporal patterns of gamma frequency oscillations tetanically induced in the rat hippocampal slice, Journal of Physiology 1997
The excitatory pyramidal cells (E-cells) now drive and synchronize the I-cells, and the I-cells gate and synchronize the E-cells. There is no need for II inhibition (but it helps a lot).
E!cells I!cells 80 0 20
0 0
50
100 time [msec]
150
200
PING is more robust than ING
When adding considerable randomness in the connectivity,
E!cells I!cells 80
0 20
0 0
50
100 time [msec]
150
200
turns into
E!cells I!cells 80
0 20
0 0
50
100 time [msec]
150
200
Where does the synchrony come from in ING and PING?
Often (not always see below), a population of neurons receiving a common inhibitory input pulse synchronizes. Simplest setting: integrate-and-re neurons subject to an exponentially decaying inhibitory synaptic input pulse. v dv = + I ge t/i v dt m v (0) = v0 [0, 1]
1
for v < 1 (ring threshold)
0.5
v
0 0
5
10
15 t [msec]
20
25
30
All neurons track a stable river (not necessarily the quasi-steady state, indicated in green in the gure).
Synchronization by a pulse of inhibition: phase plane picture
C. Brgers & N. Kopell, Synchronization in networks of excitatory and o inhibitory neurons with sparse, random connectivity, Neural Computation 2003
dv v = + I g e t/i v dt m 1 + ge t/i : m green: steady state
0.5 v w! 1
Make the equation autonomous by dening w = dv dt dw dt
= wv + I = 1 i w + 1 m
w
0.5 0.4 0.3 0.2 0
There is a single trajectory, originating in (v , w ) = (0, ), that is exponentially attracting. All trajectories exit at approximately the same w = w and therefore at approximately the same time.
The synchronizing river for theta neurons
The theta neuron (B. Ermentrout and N. Kopell, SIAM J. Appl. Math. 1986) is a simple caricature of a neuron of type I that is, a neuron that goes from rest to spiking via a saddle-node bifurcation on an invariant circle (or perhaps a homoclinic bifurcation, but in any case not a Hopf bifurcation): d = 1 cos + I ge t/I (1 + cos ) dt
J = I ge t/I
C. Brgers and N. Kopell, Eects of noisy drive on rhythms in networks of o excitatory and inhibitory neurons, Neural Computation 2005
Use of the phase plane picture to analyze synchronization by inhibitory pulses
This picture does not depend on the strength g of the inhibitory pulse, so in particular J does not depend on g . The time T between the arrival of an inhibitory pulse and the next spike is determined by the equation J = J : J = J I ge T /i = J T = i ln g I J
T = i ln g + terms independent of g The priod PING oscillation depends only logarithmically on the strength of inhibition. Therefore heterogeneity in connection strengths will not aect synchronization very much.
The eect of randomness in connectivity on PING
C. Brgers & N. Kopell, Synchronization in networks of excitatory and o inhibitory neurons with sparse, random connectivity, Neural Computation 2003
A: all-to-all coupling B: sparse, random coupling This eect comes almost entirely from the fact that dierent E-cells receive dierent total amounts of inhibition. It goes away when the number of inhibitory inputs per E-cell is xed (while connectivity is still random).
Analysis of the eect of random connectivity on PING
C. Brgers & N. Kopell, Synchronization in networks of excitatory and o inhibitory neurons with sparse, random connectivity, Neural Computation 2003
Assumption and notation: The strength of the connection from the i-th I-cell to the j-th E-cell is gIE Xij pNI where gIE > 0 is xed, and Xij = 1 with probability p, Xij = 0 with probability 1 p. The Xij are independent of each other.
Total amount of inhibitory input to a given E-cell:
NI
g=
i=1
gIE Xi pNI
Variance: Var(g ) =
2 gIE p 2 NI2 NI
Var(Xi ) =
i=1
2 gIE E(Xi2 ) E(Xi )2 = p 2 NI
2 gIE g2 E(Xi ) E(Xi )2 = 2IE p(1 p) p 2 NI p NI
Coecient of variation: Var(g ) = E(g ) 1p pNI
Now remember the formula T = i ln g + terms independent of g . If g is perturbed by g , then T is perturbed by t, with i T g g T i g . T Tg
Combining this with the formula for the coecient of variation in g , we nd: Var(T ) i E(T ) T 1p pNI
The main point is that what matters is pNI , the expected number of inhibitory inputs per E-cell, not the network size.
Where does the 40 Hz frequency come from?
Roughly speaking, the answer is: From the decay time constant of GABAA -receptor mediated inhibition. This decay time constant is often said to be around 10 ms. (Experimental data vary on this.) The time that it takes for e t/10 to decay by a factor of 10 is about 23 ms. The frequency of ING and PING does depend on external drives.
E-cells of type II
A neuron is type II if the transition from rest to spiking is via a Hopf bifurcation. As Bard and Jon explained to me this morning, a neuron with a type 2 phase response curve is likely to be a type II neuron in the sense of having a Hopf bifurcation: Resonators can be set back by excitatory input, while integrators cant. Pyramidal neurons in supercial cortical layers appear to have type 2 phase response curves:
Y. Tsubo, M. Takada, A. Reyes and T. Fukai, Layer and frequency dependencies of phase response propertis of pyramidal neurons in rat motor cortex, European Journal of Neuroscience 2007
It is therefore interesting to understand how PING works when the E-cells are type II.
PING does not usually work for E-cells of type II
C. Brgers, M. Krupa, and S. Gielen, in preparation. o
An inhibtory population spike volley creates a stable spiral (not a node). This creates a stable river, as before. As the inhibition decays, the xed point turns into a weakly repelling spiral. If the E-cell trajectories have come too close to the xed point, they are forced through sub-threshold oscillations (STOs) before they can leave. The number of STOs is sensitively dependent on initial conditions. Synchronization occurs only if inhibition is not strong enough to give rise to STOs as it decays but strong enough to create the synchronizing river. The details are more complicated. Synchronization can occur with a small number of STOs preceding each spike as long as all neurons undergo the same number of STOs on each cycle.
Example: Response of a classical Hodgkin-Huxley neuron to a strong inhibitory pulse
Classical Hodgkin-Huxley neurons are of type II.
C. Brgers, M. Krupa, and S. Gielen, in preparation. o
Can gamma oscillations synchronize over large cortical distances?
The issue is whether synchronization is prevented by conductance delays. The question is important for multiple reasons, including the widely debated binding hypothesis, according to which neuronal ensembles coding for dierent aspects of the same thing in dierent parts of the brain oscillate in synchrony at gamma frequency.
C. von der Malsburg and W. Schneider, A neural cocktail-party processor, Biological Cybernetics 1986
Synchronization of gamma oscillations over large cortical distances in vivo
There is in fact evidence that gamma oscillations do synchronize over substantial distances in the cortex, for example between primary visual cortex and extrastriate visual cortex:
A. Engel, A. Kreiter, P. Knig, and W. Singer, Synchronization of o oscillatory neuronal responses between striate and extrastriate visual cortical areas of cat, PNAS 1991
Cross-correlogram of multi-unit activity in cat primary visual cortex (area 17) and an extrastriate visual cortical area:
Synchronization of gamma oscillations over large cortical distances in vitro
R. Traub, M. Whittington, I. Stanford, and J. Jeerys, A mechanism for generation of long-range synchronous fast oscillations in the cortex, Nature 1996
Local eld potentials at two sites in CA1, separated by 4 mm:
Conduction speed in hippocampal bres is said to be about 0.5 mm/ms (Buzski et al., Hippocampus 1991), so a spatial separation of 4 a mm may translate into conduction delays on the order of 8 ms.
Possible role of inhibitory doublets in synchronization of gamma oscillations over large cortical distances
The following gure from
R. Traub, M. Whittington, I. Stanford, and J. Jeerys, A mechanism for generation of long-range synchronous fast oscillations in the cortex, Nature 1996
shows a recording from an interneuron during synchronized gamma oscillations in distant locations of a CA1 slice:
Traub et al. conjectured that long-distance synchronization involved doublets in the fast-spiking interneurons, and conrmed this by computational simulations.
Analysis of role of inhibitory doublets in synchronization of gamma oscillations over large cortical distances
B. Ermentrout and N. Kopell, Fine structure of neural spiking and synchronization in the presence of conduction delays, PNAS 1998
local circuit A
local circuit B
E
E
I
I
delayed connections
An E-cell spike in circuit A res the I-cell of circuit A immediately, that of circuit B with a delay, and vice versa. Therefore each I-cell res doublets. If B is behind, the second I-cell spike comes later in A than in B. Therefore A is held back more than B. This drives the two circuits towards synchrony.
Numerical illustration of the Ermentrout/Kopell doublet mechanism
Simulation using the model of Ermentrout and Kopell (but in a somewhat dierent parameter regime):
first circuit 50 0 V !50 !100 0 50 0 V !50 !100 0 50 100 t 150 200 50 100 t second circuit 150 200
(The delay was 5 ms here.)
Long-distance EE connections can disrupt the doublet mechanism
With weak long-distance EE connectivity, the previous example survives:
first circuit 50 0 V !50 !100 0 50 0 V !50 !100 0 50 100 t 150 200 50 100 t second circuit 150 200
but not with somewhat stronger EE connectivity (although still gEE ,dist < gEI ,dist ):
first circuit 50 0 V !50 !100 0 50 0 V !50 !100 0 50 100 150 200 50 100 t second circuit 150 200
Long-distance EE connections (without any I-cells) can synchronize if the delays are long enough
Two uncoupled E-cells (as in Ermentrout and Kopell, PNAS 1998):
first circuit 50 0 V !50 !100 0 50 0 V !50 !100 0 20 #0 $0 %0 100 20 #0 $0 t second circuit %0 100
t
synchronize when coupled with a delay of 15 ms:
first circuit 50 0 V !50 !100 0 50 0 V !50 !100 0 20 #0 $0 %0 100 20 #0 $0 t second circuit %0 100
t
or a delay of 10 ms:
first circuit 50 0 V !50 !100 0 50 0 V !50 !100 0 20 #0 $0 %0 100 20 #0 $0 t second circuit %0 100
t
but anti-synchronize with a delay of 5 ms:
first circuit 50 0 V !50 !100 0 50 0 V !50 !100 0 20 #0 $0 %0 100 20 #0 $0 t second circuit %0 100
t
For theta neurons, it is not hard to analyze this completely using the non-innitesimal PRC.
Two other models that can produce long-distance synchrony of gamma oscillations
A. Tort et al., On the formation of gamma-coherent cell assemblies by oriens lacunosum-moleculare interneurons in the hippocampus, PNAS 2007 G. Buzski et al., Circuit complexity and axon wiring economy of cortical a interneurons, Trends in Neuroscience 2004
These two models have in common that they rely on multiple classes of inhibitory interneurons, some of which are less local than others. In the model of Tort et al., long-distance synchrony arises from O-LM interneurons.
What (if anything) might gamma oscillations be useful for?
Synchrony makes excitation eective. Asynchrony makes inhibition eective. Both statements t well with the experimentally observed correlation between states of attention and gamma rhythmicity. Both statements can be made precise in multiple ways. Unfortunately, whether or not they are true depends on just how they are made precise. Intuitively, synchrony makes excitation eective because it does not leave time for positive charge entering the target neuron to leak back out, and asynchrony makes inhibition eective because it leaves no windows of opportunity. I will now give precise versions of these two statements.
For excitatory inputs, pulsed delivery is more eective than asynchronous delivery
(at least in one sense, which I am about to explain)
Consider the integrate-and-re neuron dV V = +I0 for 0 V 1, V (t + 0) = 0 if V (t 0) = 1. dt If I0 is constant, there is periodic spiking if and only if I0 > 1. The period is then I0 , T0 = ln I0 1 and the frequency is f0 = 1 . T0
One can substantially accelerate the neuron by delivering the input in periodic pulses: Replace I0 by I= k integer with T chosen so that the temporal average of I is I0 : T = 1/I0 . The frequency now becomes 1 = I0 . f= T An easy calculation shows: 1 f = > f0 1 exp(1/( f0 ))
2
(t kT ) ,
1.5 !f 1 0.5 0 0
0.5
1 !f 0
1.5
2
For inhibitory inputs, asynchronous delivery is most eective
C. Brgers and N. Kopell, Eects of noisy drive on rhythms in networks of o excitatory and inhibitory neurons, Neural Computation 2005
If a periodic inhibitory synaptic input suppresses an integrate-and-re target neuron, then the temporal average of the same input also suppresses the target neuron.
Precise formulation: Proposition: Let I I > 0, Vrev I (think of Vrev as the R, R reversal potential of an inhibitory synapse), and let g = g (t) be a function with period T . Dene V = V (t) by dV V = + I + g (t)(Vrev V ) , dt V (0) = 0 . Let g = 1 T
T
g (t) dt, and dene V = V (t) by
0
dV V = + I + g Vrev V dt V (0) = 0 . Then sup V (t) sup V (t) .
t0 t0
,
dV dt V (0)
= =
0
V
+ I + g (t)(Vrev V )
dV dt V (0)
= =
0
V
+I +g
Vrev V
sup V sup V
Proof: Assume w.l.o.g. sup V < . Then dV sup V + I + g (t) (Vrev sup V ) . dt Average over [t, t + T ]: V (t + T ) V (t) sup V + I + g (Vrev sup V ) T The right-hand side must then be 0, otherwise sup V = . This implies: V > sup V dV /dt < 0 , so V cannot go above sup V . QED
Competition among E-cells with oscillatory inhibition
C. Brgers, S. Epstein, and N. Kopell, Background gamma rhythmicity o and attention in cortical local circuits: a computational study, PNAS 2005
EW gIE stronger input gEI I gIE
EL
weaker input
If the synaptic interactions are strong and the I-cell has a low intrinsic frequency, there is a PING rhythm between EW and I, with EL silenced altogether. (W stands for winner, and L for loser.) EW recruits enough inhibition to suppress EL .
Competition among E-cells with asynchronous (constant) inhibition
EW gIE stronger input I I I I I
EL
weaker input I
Assume that the I-cells spike asynchronously (for some reason not discussed here), and that there are many of them, so each of the two E-cells receives constant synaptic inhibition. It is possible to choose gIE so that EL is silenced, but EW is not.
Competition by rhythmic inhibition is more eective than competition by asynchronous inhibition
In both scenarios, choose gIE just barely strong enough to suppress EL . The inhibition aects EW as well. How much is EW slowed down?
with rhythmic inhibition
EW
EL
with asynchronous inhibition
In the oscillatory network, the inhibition is recruited when EW has just spiked, and EL is about to spike. The timing is favorable to EW , and unfavorable to EL . With rhythmic inhibition, EW wins at less cost to itself than with asynchronous inhibition.
The stimulus competition eect
This is an surprising eect observed in experiments with monkeys. In the next slides, I will suggest an explanation of it that uses gamma oscillations. Here is the eect:
J. Reynolds, L. Chelazzi, and R. Desimone, Competitive mechanisms subserve attention in macaque areas V2 and V4, Journal of Neuroscience 1999
The presence of a poor yet weakly excitatory stimulus reduces the response to a good stimulus.
The explanation of the stimulus competition eect by Reynolds et al.
J. Reynolds, L. Chelazzi, and R. Desimone, Compeitive mechnisms subserve attention in macaque areas V2 and V4, Journal of Neuroscience 1999
Reynolds et al. use the simplest possible rate model: dY Y = + we (Ye Y ) wi Y dt with Y = ring rate. For we > 0 and wi > 0, the steady state Yeq = we Ye 1/ + we + wi
is positive. So any stimulus with we > 0 and wi > 0 is excitatory when presented by itself. But if one raises we a little, and wi a lot, of course Yeq decreases. So a stimulus can be excitatory by itself, but suppressive when added to another stimulus.
The explanation of the stimulus competition eect by Reynolds et al. fails as soon as spiking is introduced
Change the model of Reynolds et merely al. by writing V instead of Y , interpreting V as membrane potential, and introducing a reset condition: dV V = + we (Ve V ) wi V dt V (t + 0) = 0 if V (t 0) = 1 Proposition: In the above model, a stimulus that is excitatory by itself (that is, a stimulus giving rise to a positive spiking frequency) cannot be suppressive (lower the spiking frequency) when added to another stimulus. So there is no stimulus competition in this model.
C. Brgers, S. Epstein, and N. Kopell, Gamma oscillations mediate o stimulus competition and attentional selection in a cortical network model, submitted for publication (2008)
An explanation of the stimulus competition eect using gamma oscillations
C. Brgers, S. Epstein, and N. Kopell, Gamma oscillations mediate o stimulus competition and attentional selection in a cortical network model, submitted for publication (2008)
When only one stimulus is presented, inhibition oscillates at gamma frequency, thereby allowing even those neurons that only receive weak (and stochastically uctuating) excitation to spike some of the time. When multiple stimuli are presented, there is so much drive to the I-cells that the gamma oscillation is abolished. A constant curtain of inhibition descends upon the whole local network.
Numerical illustration of our explanation of the stimulus competition eect
Responses to individual stimuli:
40 inh. cond. I!cells
A
20 1 0
0.5
B
100 200 time [ms]
300
0 0 40 20 0 0
100 200 time [ms]
300
160 80 1 160 80 1 freq [Hz] E!cells freq [Hz] E!cells
C
D
time [ms]
80 neuronal index
160
E
40 20 0 0 80 neuronal index
F
time [ms]
160
Response to both stimuli together:
40 inh. cond. I!cells
A
20 1 0
0.5
B
100 200 time [ms]
300
0 0 50 freq [Hz]
100 200 time [ms]
300
160 E!cells
C
80 1
D
time [ms]
0 0
80 neuronal index
160
Beta rhythms (1230 Hz)
Gamma-beta transition in a rat hippocampal slice
M. Whittington, R. Traub, H. Faulkner, I. Sanford and J. Jeerys, Recurrent excitatory postsynaptic potentials induced by synchronized fast cortical oscillations, PNAS 1997
100 ms
This is an extracellular eld potential recorded in a rat hippocampal slice (CA1, stratum pyramidale) following tetanic stimulation. There is a spontaneous transition from gamma to beta oscillation.
Gamma-beta transition in human brains
C. Haenschel, T. Baldenweg, R. Croft, M. Whittington, and J. Gruzlier, Gamma and beta frequency oscillations in response to novel auditory stimuli: A comparison of human electroencephalogram (EEG) data with in vitro models, PNAS 2000
gamma (30-50 Hz) slow beta (12-20 Hz)
stimulus
stimulus
These are band-passed ltered EEG signals in response to an auditory stimulus. An initial stimulus-locked gamma burst shifts to a slow beta oscillation.
Beat-skipping beta rhythms
This form of beta oscillation was described for instance in
M. Whittington, R. Traub, N. Kopell, B. Ermentrout, and E. Buhl, Inhibition-based rhythms: experimental and mathematical observations on network dynamics, International Journal on Psychophysiology 2000.
Start with a network in which PING occurs, but I-cells driven strongly, gII large, so ING would be possible without the E-cells. Then a strong depolarization-induced hyperpolarizing current (such as an AHP current) can produce this pattern:
E!cell 50 0 !50 !100 0 50 0 !50 !100 0 100 200 300 time [ms] 400 500 100 200 I!cell 300 400 500
E-cell spikes at beta, I-cell spikes at gamma
Gamma-beta transition in a two-cell network
gm = maximum conductance of AHP-current
g =0
m
g =0.06
m
0
100
200 300 gm=0.02
400
0
100
200 300 gm=0.08
400
0
100
200 300 gm=0.04
400
0
100
200 300 gm=0.1
400
0
100
200
300
400
0
100
200
300
400
The range of values of gm that yield beat-skipping beta is fairly large.
Dierent E-cells may skip dierent beats
With an AHP current for which there is beat-skipping beta in a two-cell network, this is what happens without E synapses when there are 10 E-cells:
E!cells 10
0
I!cell 0
100
200 time [msec]
300
400
A small amount of EE conductance brings the E-cells together, restoring the beta rhythm
E!cells 10
0
I!cell 0
100
200 time [msec]
300
400
R. Traub et al., On the mechanism of the frequency shift in neuronal oscillations induced in rat hippocampal slices by tetanic stimulation, Journal of Neuroscience 1999 N. Kopell, B. Ermentrout, M. Whittington, and R. Traub, Gamma rhythms and beta rhythms have dierent synchronization properties, PNAS 1999
Long-distance synchronization of beta rhythms
N. Kopell, B. Ermentrout, M. A. Whittington, and R. D. Traub, Gamma rhythms and beta rhythms have dierent synchronization properties, PNAS 2000
The beat-skipping beta rhythms can synchronize using the doublet mechanism. There are I-cell doublets on the gamma cycles on which the E-cells spike, and I-cell singlets on the gamma cycles on which the E-cells are silent. Synchrony is achieved with longer delays than for gamma oscillations.
Movement onset and beta rhythms
J. Sanes and J. Donoghue, Oscillations in local eld potentials of the primate motor cortex during voluntary movement, PNAS 1993
1 sec
movement onset
These are local eld potentials measured from various locations in motor cortex. Beta rhythms precede movement onset, but are greatly reduced during movement. Some of the recording sites are several millimeters apart, yet coherent. These and many similar results suggest that (some) beta rhythms reect inhibition of the motor cortex.
Pathological beta rhythms and Parkinsons disease
In Parkinsons disease, beta power increases in extracellular and eld potential recordings from the subthalamic nucleus. (R. Levy et
al., Dependence of subthalamic nucleus oscillations on movement and dopamine in Parkinsons disease, Brain 2002)
Levodopa (used in the treatment of Parkinsons disease to increases dopamine levels) has the eect of reducing beta oscillations in the subthalamic nucleus and internal globus pallidus.
(P. Brown et al., Journal of Neuroscience 2001)
There appears to be a connection between pathological beta rhythmicity in the basal ganglia and Parkinsons disease.
Pathological slower rhythms and Parkinsons disease
Pathological slower rhythms (48 Hz, which is the frequency of Parkinsonian tremor) have also been reported in Parkinsons disease. For example:
H. Bergman, T. Wichman, B. Karmon, and M. DeLong, The primate subthalamic nucleus. II. Neuronal activity in the MPTP model of Parkinsonism, Journal of Neurophysiology 1994. A. Raz, E. Vaadia, and H. Bergman, Firing patterns and correlations of spontaneous discharge of pallidal neurons in the normal and the tremulous 1-methyl-4-phenyl-1,2,3,6-tetrahydropyridine vervet model of Parkinsonism, Journal of Neuroscience 2000
The connection between pathological slower rhythms and Parkinsons disease was modeled by Rubin and Terman:
J. Rubin and D. Terman, High frequency stimulation of the subthalamic nucleus eliminates pathological thalamic rhythmicity in a computational model, Journal of Computational Neuroscience 2004
Alpha rhythms (712 Hz)
Places in the brain where alpha frequency rhythms are found
in the occipital lobe (which contains the visual cortex), thought to arise there in layer V (which contains large pyramidal cells that project to subcortical areas) in motor cortex (mu rhythm) in the thalamus during early stages of sleep (sleep spindles)
Modeling alpha rhythms
Historically, the alpha rhythm was rst modeled by Norbert Wiener1 (Nonlinear Problems in Random Theory, MIT Press 1958). Models of alpha rhythms typically involve various slow currents: the h-current, a current that is carried by sodium and potassium, on balance depolarizing, slowly activated by hyperpolarization (hence h) the T-current, a current carried by calcium and hence strongly depolarizing, turns on only if rst de-inactivated by prolonged hyperpolarization, then activated by depolarization AHP-currents, carried by potassium and hence hyperpolarizing, activated by depolarization
1
graduated from Tufts University with a B.A. in mathematics in 1909, at age 14
Model of cortical alpha rhythms by Jones, Pinto, Kaper, and Kopell
S. Jones, D. Pinto, T. Kaper, and N. Kopell, Alpha-frequency rhythms desynchronize over long cortical distances: a modeling study, J. Comp. Neurosci. 2000 D. Pinto, S. Jones, T. Kaper, and N. Kopell, Analysis of state-dependent transitions in frequency and long-distance coordination in a model oscillatory cortical circuit, J. Comp. Neurosci. 2003
The focus of the above papers is on synchronization over distances. I will only discuss their model of the local alpha rhythm here.
In the model of Pinto et al., isolated E-cells spike at alpha frequency
50 0
E
V !50 !100 0 200 400 600 800 1000
The frequency of the isolated E-cell is set by the interaction of the h-, T- and AHP-currents. The following table shows how the frequency changes when various parameters are raised by 10%: h gh T ,inact T ,act gT AHP gAHP 1.8% +2.6% +1.6% 2.4% +14% 6.0% 8.9% So all these parameters matter in setting the frequency.
In the local circuit of Pinto et al, the I-cells hardly aect the frequency
Coupling the E-cell to an I-cell with AMPA and GABAA -receptor mediated synapses does not have any signicant eect on the oscillation:
50 0
E
E!cell !50 !100 0
100
200
300
400
500
600
700
800
900
1000
50
I
I!cell
0
!50
!100 0
100
200
300
400
500
600
700
800
900
1000
In the model of Pinto et al., two E-cells synchronize at alpha frequency when there is EE-conductance:
50
E
E
E!cells
0
!50
!100 0
100
200
300
400
500
600
700
800
900
1000
Again the I-cell plays no role in the local circuit:
50 E!cells (gren and black) 0
!50
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With an I-cell, but without EE-condcutance, two E-cells in the model of Pinto et al. anti-synchronize:
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Sleep spindles
Sleep spindles are seen in the EEG during the early stage of sleep.
M. Steriade and D. McCormick and T. Sejnowski, Thalamocortical oscillations in the sleeping and aroused brain, Science 1993
This is a local eld potential, recorded in the reticular nucleus of the thalamus of a cat. Spindle episodes occur every 3 seconds or so, and last for about 1 second.
Within each spindle episode, there is an alpha oscillation
This is a gure from X.-J. Wang and J. Rinzel, Neuroscience 1993, showing the simulated voltage trace of a neuron in the reticular thalamic nucleus oscillating at alpha frequency ( 10 Hz) during a spindle episode.
Rapid spike bursts ride on the peaks of the alpha frequency pacemaker oscillation within each spindle episode
The gure on the right is from M. Steriade and I. Timofeev, Neuron 2003, and shows an intracellularly recorded voltage train of a cat thalamocortical neuron. In the simulation of Wang and Rinzel, spiking currents were omitted.
Models of sleep spindles
Sleep spindles can be generated by inhibitory cells in the reticular nucleus of the thalamus (RE cells) alone:
X.-J. Wang and J. Rinzel, Spindle rhythmicity in the reticularis thalami nucleus: synchronization among mutually inhibitory neurons, Neuroscience 1993
They can also be generated by the interplay between RE cells and thalmocortical relay (TC) cells:
D. Golomb, X.-J. Wang, and J. Rinzel, Synchronization properties of spindle oscillations in a thalamic reticular nucleus model, Journal of Neurophysiology 1994
The waxing and waning of spindles is thought to be the result of calcium entry into the cell (the T-current), which strengthens the h-current until de-inactivation of the T-current is made impossible.
A. Destexhe, A. Babloyantz, and T. Sejnowski, Ionic mechanisms for intrinsic slow oscillations in thalamic relay neurons, Biophysical Journal 1993
Speculations on the function of spindle and delta oscillations during sleep
M. Steriade and I. Timofeev, Neuronal plasticity in thalamocortical networks during sleep and waking oscillations (Review), Neuron 2003
Cortical disconnection: Oscillations entrain thalamocortical relay neurons and prevent them from tarnsmitting sensory information to the cortex. Consolidation of memory traces: Rhythmic and synchronized spike bursts of thalamic neurons depolarize the dendrites of neocortical neurons. This is associated with entry of Ca2+ into the neurons, which may support synaptic plasticity of excitatory synapses in cortex.
Theta rhythms (3 7 Hz)
Some examples of theta rhythms in the brain
very strong in hippocampus and entorhinal cortex of rodents during learning and memory retrival bursts of hippocampal theta in humans during REM sleep
J. Cantero et al., Sleep-dependent theta oscillations in the human hippocampus and neocortex, Journal of Neuroscience 2003
these are intracranial EEG traces from patients with epilepsy (but probably show non-pathological activity) in hippocampus, basal temporal lobe, and frontal cortex during transition from sleep to wakefulness frontal midline theta is pronounced during states of sustained attention
K. Inanaga, Frontal midline theta rhythm and mental activity (Review Article), Psychiatry and Clinical Neurosciences 1998
A model of a hippocampal theta rhythm
H. Rotstein et al., Slow and fast inhibition and an h-current interact to create a theta rhythm in a model CA1 interneuron network, Journal of Neurophysiology 2005
In this model, the theta rhythm arises from the interaction of two populations of neurons: fast-spiking interneurons (I-cells) (thought to be the basis of gamma and cycle-skipping beta oscillations) oriens-lacunosum moleculare interneurons (O-cells)
I-cells vs. O-cells in the model of Rotstein et al.
I-cells are assumed to have no currents beyond the standard Hodgkin-Huxley spiking currents. The decay time constant of inhibition originating from I-cells is assumed to be 10 ms. O-cells are assumed to have additional currents, most importantly an h-current (hyperpolarization-activated, non-inactivating, depolarizing). The decay time constant of inhibition originating from O-cells is assumed to be 20 ms. .
Interaction of I- and O-cells generates theta rhythm
IO inhibition activates h-current, is therefore excitatory. OI inhibition is truly inhibitory, and synchronizing as in PING (but with a longer period since it decays more slowly). The frequency is determined primarily by the decay time constant of OI inhibition and by the dynamics of the h-current.
The delta rhythm (0.5 3 Hz)
The delta rhythm
Slow (0.53 Hz), occurs during deep sleep. Thalamus involved in generating it. During deep sleep, there is also an even slower rhythm (< 1 Hz) in the cortex. Simulation from
D. Terman and A. Bose and N. Kopell, Functional reorganization in thalamocortical networks: Transition between spindling and delta sleep rhythms, PNAS 1996:
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The plot shows average voltages of thalamocortical (TC) cells. The delta oscillations are triggered (in the model) by input from the slow cortical oscillation, which briey activates RE cells, giving rise to slow, GABAB -receptor mediated inhibition of the TC cells. This de-inactivates the T-current, which drives the oscillations.
Very fast oscillations (90 200 Hz)
Very fast oscillations ( 200 Hz)
Very fast oscillations (VFOs) are seen in hippocampus,
G. Buzski et al., High-frequency network oscillation in the hippocampus, a Science 1992 A. Draguhn, R. Traub, D. Schmitz, and J. Jeerys, Electrical coupling underlies high-frequency oscillations in the hippocampus in vitro, Nature 1998
in neocortex during slow-wave sleep:
F. Grenier, I. Timofeev, and M. Steriade, Focal synchronization of ripples (80200 Hz) in neocortex and their neuronal correlates, Journal of Neurophysiology 2001
and also preceding/accompanying epileptic activity:
F. Grenier, I. Timofeev, and M. Steriade, Neocortical very fast oscillations (ripples, 80200 Hz) during seizures: intracellular correlates, Journal of Neurophysiology 2003
Evidence pointing towards a central role of gap junctions in the generation of hippocampal VFOs
Gap junction blockers reduce or abolish VFOs.
A. Draguhn, R. Traub, D. Schmitz, and J. Jeerys, Electrical coupling underlies high-frequency oscillations in the hippocampus in vitro, Nature 1998 M. Jones et al., Intracellular correlates of fast (> 200 Hz) electrical oscillations in rat somatosensory cortex, Journal of Neurophysiology 2000
VFOs persist after blocking GABAA -recptors with bicuculline, and also with ionotropic glutamate receptor antagonists.
A. Draguhn, R. Traub, D. Schmitz, and J. Jeerys, Electrical coupling underlies high-frequency oscillations in the hippocampus in vitro, Nature 1998
VFOs in networks of principal cells connected by axo-axonal gap junctions
Draguhn et al. hypothesized that hippocampal VFOs are generated by networks of principal cells with axo-axonal gap junctions.
A. Draguhn, R. Traub, D. Schmitz, and J. Jeerys, Electrical coupling underlies high-frequency oscillations in the hippocampus in vitro, Nature 1998
Computer simulations conrmed this possibility.
R. Traub, D. Schmitz, J. Jeerys, and A. Draguhn, High-frequency population oscillations are predicted to occur in hippocampal pyramidal neuronal networks interconncted by axoaxonal gap junctions, Neuroscience 1999
Noise-driven target patterns in networks of principal cells connected by gap junctions
T. Lewis and J. Rinzel, Self-organized synchronous oscillations in a network of excitable cells coupled by gap junctions, Network 2000
cellular automaton model
One cycle of the oscillation: A ring-shaped wave of activity passes through the network, triggered by a random excitatory event. When another random event triggers the next wave, a new cycle starts.
Self-sustaining spiral waves in networks of principal cells connected by gap junctions
E. Munro, The axonal plexus: a description of the behavior of a network of axons connected by gap junctions, Ph. D. Thesis, Tufts University, May 2008
modiciation of the Lewis/Rinzel cellular automaton in which dierent cells are allowed to have (slightly) dierent refractory periods
Munro also showed that this sort of persistent VFOs occur in the full model of Traub et al., for intermediate coupling strengths: Strong enough for signal transmission to work most of the time, but not so strong that it works every time.
Neocortical VFOs may involve fast-spiking interneurons
F. Grenier, I. Timofeev, and M. Steriade, Focal synchronization of ripples (80200 Hz) in neocortex and their neuronal correlates, Journal of Neurophysiology 2001
Grenier, Timofeev, and Steriade showed that fast-spiking interneurons spike phase-locked with ripples. They hypothesize that the mechanism generating VFOs in neocortex involves fast-spiking interneurons.
Possible functional role of VFOs
VFOs reach their strongest amplitudes during slow-wave sleep.
F. Grenier, I. Timofeev, and M. Steriade, Focal synchronization of ripples (80200 Hz) in neocortex and their neuronal correlates, Journal of Neurophysiology 2001
Slow-wave sleep is believed to play a role in consolidation of memory traces.
M. Steriade, Impact of network acivities on neuronal properties in cortico-thalamic systems, Journal of Neurophysiology 2001
Hippocampal ripples may constitute a replay at a faster time scale of ring sequences coding for important events
Z. Nadsdy et al., Relay and time compression of recurring spike a sequences in the hippocampus, Journal of Neuroscience 1999.
Precise timing of ring is thought to be important for plasticity. Grenier et al. therefore hypothesized that VFOs may play a role in consolidation of memory traces during slow-wave sleep.
Concluding comments
I selected topics that I have thought about, or at least read about. The subject is huge, and I omitted a great number of important topics (nested theta-gamma rhythms, functional signicance of the theta rhythm, long-distance synchronization of slower rhythms, and many, many others) for no reason other than my own ignorance. Thank you very much for listening for so long!