The neuron's response at extended timescales

0.89 MB·English

by  Daniel Soudry

The neuron’s response at extended timescales
Daniel Soudry and Ron Meir
Department of Electrical Engineering, the Laboratory for Network Biology Research, Technion 32000, Haifa , Israel
Many systems are modulated by unknown slow processes. This hinders analysis in highly non-
linear systems, such as excitable systems. We show that for such systems, if the input matches
the sparse “spiky” nature of the output, the spiking input-output relation can be derived. We use
thisrelationtoreproduceandinterprettheirregularandcomplex1/f responseobservedinisolated
neuronsstimulatedoverdays. Wedecomposetheneuronalresponseintocontributionsfromitslong
history of internal noise and its “short” (few minutes) history of inputs, quantifying memory, noise
and stability.
PACSnumbers: 87.19.ll,87.85.dm,87.19.lc,87.19.lr,87.10.Mn,82.20.Fd
3
1
Many models, especially in biology, are accurate only
0
below a certain timescale - due to the existence of addi-
2
Stimuli
tionalslowprocesses. Iftheseslowprocessesarenotwell
n
characterized,itmaybehardtopredicthowtheywillaf- Response
a
J fectthedynamicsatlongertimescales. Thisisespecially

1 trueifthedynamicsarefarfromequilibrium,highlynon-
1 linearandcontainfeedback,aregimewhereexcitabilityis Kinetic noise - "State” -AP generation noise
atypicaldynamicalphenomenon[1]. Amongmanytypes Input - -Output
]
C of excitable systems (e.g., [1] and references therein), a Kinetics “Weights”
Legend:
neuron is a prototypical example - where Action Poten-
N
AP affecting
tials (AP - a stereotypical voltage “spike”) are generated
kinetics
.
o in response to stimulation [2]. AP generation is indeed
i affected by many slow processes [3] - with new processes
b FIG. 1: Mathematical analysis - schematic summary
- beingdiscoveredatanexplosiverate[4–6]. Thismayen- A Our aim: to find the I/O relation between inter-stimulus
q
tail a complex stochastic and history-dependent Input- intervals (T ) and Action Potential (AP) occurrences (Y ) -
[ m m
Output (I/O) relation, on multiple timescales [7–9]. In for a general biophysical neuron model (Eq. 1-3). B An AP

1 general, it is hard to identify, simulate or analyze such “occurred” if the voltage V crossed a threshold V following
th
v an I/O due to the large number of processes which are the stimulus. We assume “sparse” stimulations, so Tm (cid:29)
1 τ . C Main result: conversion of a complex biophysical
unknown or lacking known parameters. AP
3 neuron model to a simple linear model with feedback (Eqs.
6 We find the situation simplifies considerably if we 6-7), where the parameters (F,d,a and w) are linked to the
2 use(experimentallyrelevant[10–14])sparsespikeinputs,
biophysical parameters of the full model (Eqs. 1-3).
. similartothetypicaloutputoftheneuron(Fig. 1A&B).
1
0 We derive, for a general biophysical stochastic neuron
3 model (Eqs. 1-3) with a few assumptions, a concise de- physical model, showing that slow processes span a wide
1 scription for the I/O (Eqs. 6-7) based on biophysically range of timescales - with slower processes being “nois-
:
v meaningful parameters. This I/O is well described by ier”, due to low ion channel population numbers. The
i an ‘engineering-style’ block diagram with feedback (Fig. model suggests the 1/fα statistics of the response orig-
X
1C), which can be used to decompose the effects of noise inates from the long history of internal noise, while in-
r
a andinputontheresponse. Beyondtheconceptuallucid- putfluctuationsonlyaffecttheresponseona(relatively)
ity,suchalinearI/Oallowstheutilizationofwellknown short timescale of a few minutes.
statistical tools to derive all second order statistics, con-
struct linear optimal estimators and perform parameter Full Model The voltage dynamics of an isopoten-
identification. Theseresultsholdnumerically,evensome- tialneuronaredeterminedbyionchannels,proteinpores
times when our assumptions break down. which change conformations stochastically with voltage-
We demonstrate the utility of our results on recent dependent rates [16]. On the population level, such dy-
experiments [13] where synaptically isolated individual namics are generically very well described by models of
neurons, from rat cortical culture, were stimulated with the form [17–19]
extra-cellular sparse current pulses for a unprecedented
duration of days. The neurons exhibited 1/fα statistics
V˙ = f(V,r,s,I(t)) (1)
[15], responding in a complex and irregular manner from
seconds to days. Using our results, we are able to repro- r˙ = Ar(V)r+Br(V,r)ξr (2)
duce and analyze the origins of this behavior in a bio- s˙ = A (V)s+B (V,s)ξ (3)
s s s
2
with voltage V, stimulation current I(t), rapid vari- T A(Y ,T )). Note that this simplified linear discrete
m m m
ables r (e.g., m,n,h in the Hodgkin-Huxley (HH) model time map has far fewer parameters than the full model,
[2]),slowvariabless(e.g.,slowsodiuminactivation[20]), since it is written explicitly only using the averaged mi-
ratematricesA , whitenoiseprocessesξ (withzero croscopicratesofs(throughA andD), populationsizes
r/s r/s
meanandunitvariance),andmatricesB whichcanbe (through D), the probability to generate an AP given s,
r/s
written explicitly using the rates and ion channel num- p (s),andtherelevanttimescales. Thiseffectivemodel
AP
bers[21](D=BB isthediffusionmatrix[21,22]). For exposesthelargedegeneracyintheparametersofthefull
(cid:62)
simplicity, we assumed r and s are not coupled directly, modelandleadstosignificantlyreducedsimulationtimes
but this is non-essential [23, 24]. The parameter space and mathematical tractability.
can be constrained [19], since we consider here only ex- Linearization Intrinsicionchannelnoisecanbeex-
citable,non-oscillatory neuronswhichdonotfiresponta- ploited to linearize the neuronal dynamics, rendering
neouslyandwhichhaveasinglerestingstate-ascommon it more tractable than the (less realistic) noiseless case
for cortical cells, e.g., [13]. Such biophysical neuronal [19]. Suppose that T has stationary statistics with
m
{ }
models(Eqs. 1-3)aregenerallycomplexnon-linearmod- mean T so that τ T τ with high probabil-
AP m s
els, containing many variables and unknown parameters ity. Sinc∗e s is slow and(cid:28)AP gen(cid:28)eration is rather noisy in
(sometimes ranging in the hundreds [25, 26]), not all of this regime [19] (so p (s ) is slowly varying [24]), we
AP m
which can be identified [27]. Therefore, such models are assume a stable excitability fixed point s exists, so per-
notoriously difficult to tune, highly susceptible to over- turbations ˆs = s s are small and w∗e can linearize
m m
fitting and computationally expensive [28–30]. Also, the p (s ) p +w ˆs−.∗The mean AP firing rate can be
AP m (cid:62) m
high non-linearity usually prevents exact mathematical found self≈con∗sistently (and rather accurately, Fig. 2A)
analysis of such models at their full level of complexity from the location of the fixed point s
[31]. ∗
Model reduction However, much of the complexity Ym =p =pAP(s (p ,T )) , (5)
(cid:104) (cid:105) ∗ ∗ ∗ ∗
insuchmodelscanbeovercomeunderawelldefinedand
while the perturbations around the fixed point are de-
experimentally relevant settings [10–14], if we use sparse
scribed by the linear system
inputs, similar in nature to the spikes commonly pro-
duced by the neuron. This is done by “averaging out”
ˆs = Fˆs +dTˆ +aYˆ +n , (6)
Eqs. 1-3 using similar methods to those in [19]. Specifi- m+1 m m m m
cally,supposeI(t)isapulsetrainarrivingattimes tm Yˆm = w(cid:62)ˆsm+em (7)
{ }
(Fig. 1A,top),soT =t t τ withτ being
m m+1 m AP AP (cid:10) (cid:11)
− (cid:29) with F = I+T A(p ,T ), n n = T D(p ,T ,s ),
the timescale of an AP (Fig. 1B). Our aim is to describe m (cid:62)m
e is a white n∗oise p∗roc∗ess (other param∗eters∗in∗[37∗]).
theAPoccurrencesY ,whereY =1ifanAPoccurred m
m m
This linear I/O, which contains feedback from the ‘out-
immediately after the m-th stimulation, and 0 otherwise
put’ Yˆ to the state variable ˆs (Fig. 1C), can be very
(Fig. 1A, bottom). To do so, we need to integrate Eqs. m m
helpfulmathematicallyandallitsparametersaredirectly
1-3 between t and t . Since T τ the rapid
m m+1 m AP
(cid:29) related to well motivated biophysical variables. More-
AP generation dynamics of (V,r) relax to a steady state
over,Thisformulationmakesitnowpossibletoconstruct
beforet . Therefore,theneuronAP“remembers” any
m+1
optimal linear estimators for Y and s [32] (Fig. 2B),
history before t only through s = s(t ). Given s , m m
m m m m
performparameteridentification, andusestandardtools
theresponseofthefastvariables(V,r)tothem-thstim-
[22]tofindallsecondorderstatisticsinthesystem, such
ulation spike will determine the probability to generate
as correlations or Power Spectral Densities (PSD). For
anAP.Thisprobability,p (s),collapsesalltherelevant
AP
example, the PSD for Y is
informationfromEqs. 1-2,andcanbefoundnumerically m
fromthepulseresponseofEqs. 1-2withsheldfixed[24]. (cid:0) (cid:1)
S (f) = w H (f) D(p ,T ,s )+dd S (f) H ( f)w
In order to integrate the remaining Eq. 3 we define the Y (cid:62) c ∗ ∗ ∗ (cid:62) T (cid:62)c −
(cid:12) (cid:12)2
averaged rate matrix + T p (1 p )(cid:12)1+w(cid:62)Hc(f)a(cid:12) (8)
∗ ∗ − ∗
A(Ym,Tm) += τ(cid:0)A1P−Tτm−A1P(TYm−m1A(cid:1)+A+0,(1−Ym)A−) wfdihff(cid:28)eerreTen∗−Ht1.cpa(Afrga)ami=net,en(cid:0)r2osπtwefiitllh−egAelnare(gprae∗t,deTe∗tg)hee−nesTraa∗m−c1yeahPweS(cid:62)rDe(cid:1).−-O1m,tahfnoeyrr
where A ,A and A are the averages of A during an
+ 0 s immediately derivable statistics are S (f), the cross-
APresponse,−afailedAPresponseandrest, respectively. s
PSDs S (f),S (f),S (f) and also the respective
Assuming T τ , we obtain, to first order sT sY TY
m (cid:28) s correlations. Our exact results agree very well with
the numerical solution of Eqs. 1-3 (Fig 2C-D), even in
s =s +T A(Y ,T )s +n . (4)
m+1 m m m m m m
somecaseswheretheunderlyingassumptionsdonothold
where n is a white noise process with zero mean (specifically if T τ and s is unstable [24]).
m m s
and variance T D(Y ,T ,s ) (defined similarly to ∼ ∗
m m m m
3
A 1 B x 10−3 (through the variable a, see Fig. 4C), comparison with
Simulation Simulation Estimate experiment implies that the observed f α dependence
−
0.8 Approximation
1 was not generated in this way, since it exists even near
0.6 the critical stimulation frequency, where p 1,w 0.
p∗0.4 ˆs 0 Therefore, the eigenvalues of A(p ,T ), the∗a→verage→rate
0.2 −1 matrix, must span a large range o∗f (∗inverse) timescales.
R2=0.88 However, the existence of this range is not sufficient - we
0
0 10 T−201[Hz3]0 40 270 27t5im2e80[sec2]85 290 also require [15] ci ∝λ−i α so that
∗ ˆ
C D
dλλ1 α
S[sec]Y1100−−32 MSAipampp.rox. S[sec]s1100−−86 Map AronbaulsytziwnagyES(qiYn.d(8fep)we∼endfienndt(2toπhffaT)t2w)−+,etλch2aant∝gisfen−ceoαrna.stiesttehnitswinitha
SAipmp.rox. otherexperimentalobservation∗s,byhavingascalingrela-
10−410−4 10−3 10−2 10−1 100 101 10−1010−4 10−3 10−2 10−1 100 101 tioninNi,thenumberofthecorrespondingionchannels
f [Hz] f [Hz] (affecting D(p ,T ,s ), the variance of n in Eq. 6).
m
FIG.2: Comparing the mathematical results with the Specifically, by∗set∗tin∗g Ni ∝ λ−i α, implying that slower
processes are noisier. To fit a specific model consider
numerical simulation of Eqs. 1-3 for the stochastic
the stochastic HHS model which augments the basic HH
HH model with slow sodium inactivation [19, 24] with
T = T = 50ms. A Firing probability p (cid:0)T−1(cid:1) (Eq. 5) model with a slow sodium inactivation variable [19, 24].
m ∗ ∗ ∗
for different currents (I = 7.5,7.7,7.9,8.1,8.3 µA from bot- We extend this model by replacing the sodium inactiva-
tom to top). B Optima0l linear estimation of sˆ. C-D Power tion variable s with (cid:80)M s /M, where s is identical to
i=1 i 1
spectral densities SY (f) and Ss(f). ‘Map’ is a (104 faster) s, and for s M the rates scale as (cid:15)i and the channel
simulation of Eq. 4 together with p (s ), while ‘Approx’ { i}i=2
AP m numbers scale as N (cid:15)αi. In order to obtain α = 1.4
is the analytical expressions (e.g., SY is given by Eq. 8). for the duration ofith∝e experiment we require (cid:15) = 0.2
I =7.9µA in B-D. For more complex models see [24].
0
and M = 5. This reproduces well the observed scaling
relations (Fig. 3).
Modeling the Experiment Next, we use the ex- Predictions First, we consider the duration of the
pression for the PSD (Eq. 8) to find conditions un- neuronal memory. To quantify this more precisely, we
der which the experimental results of [13] can be repro- note that in the frequency domain we can use spectral
duced, given our assumptions. We will show that kinetic factorization [32] and write the response of the linear
processes must possess a large range of slow timescales, system (Eqs. 6-7) as
where the number of relevant ion channels with each
Yˆ (f)=H (f)Tˆ(f)+H (f)ν(f) (9)
timescale scales with an exponent α. Fitting a specific signal noise
model we reproduce the experimental results in Fig. 3.
with H (f) = w H (f)d and ν(f) is the Fourier
Previous work [19] used a stochastic HH model with signal (cid:62) c
transform of a zero mean white noise representing in-
slow sodium inactivation to reproduce the experimental
ternal neural noise. In our model, the f α behavior is
results of [13] up to a timescale of minutes. However, −
generated by fluctuations in internal neuronal noise, so
thismodelcannotexplaindynamicsonlongertimescales
H (f) f α. However, H (f) c for f 0
(1F0ig2.H7zC,)w.iSthpeαcifica1ll.y4,wwheerneqTuire=STY (,fa)s∼inf[−1α3].foIrffth<e sinncoieseit is n∼ot a−ffected by Ni. Tshiginsaelntail∼s that the→neu-
tot−al number of i≈on channel stamtes of∗all channel types is ron possesses a long memory for its intrinsic fluctuations
but a “finite memory” of its input - i.e. perturbations in
finite, then we can decompose Eq. 8
Y due to perturbations in T will decay exponentially
m m
with a finite timescale. Specifically, in the fitted model,
M
SY (f)=c0+(cid:88)(2πfc)i2λi+λ2 , Ha stiigmnaels(cfa)lehoafs the10s0hsaepce. oTfhaissicmopulledlaolwsopbaesspfirlotbeerdweitxh-
i=1 i ∼
perimentally directly by applying a sinusoidal T input
m
where c are some (derivable) constants and the poles λ
i i
(cid:88)
are solutions of the characteristic equation T =T + T sin(2πf T m) , (10)
m 1 i
∗ ∗
(cid:12) (cid:12) i
(cid:12)λI A(p ,T ) T−1aw(cid:62)(cid:12)=0.
− ∗ ∗ − ∗ since the linear response of the model will generate a
In order to approximate a PSD of the form f α, we re- direct probe of H (f) (Fig. 4A). Note that this pre-
− signal
quire the poles to cover a large range [15]. Though the diction of input memory timescale of 100sec is valid
∼
T 1aw feedbacktermcantunethelocationofthepoles only for spiking input, in the context of our model. In
− (cid:62)
∗
4
A 2 B1.5 A 102 B 1
RateFluct.[Hz]−−01210 10 20 103s0 N4o0r3m0asl5iz0edt1i6m000es 70 30800s 90 100 CV0.015 100 T101[secEM]1x0op2deerlime1n0t3 H(f)[Hz]signal||111000−011 SAipmpurolaxt.ion 2S[sec]YT||00..027.555 EAxppperorixm.ent
C D E
103 102 101 10−120−5 10−4 10−3 f1[0H−2z] 10−1 100 101 0 0.01 0.02 f0[.H03z] 0.04 0.05 0.06
PSD101 DFA101 FF100 FIG. 4: Input memory in fitted model. (A)
10−1 10−1 10−1 ‘Approx’:|Hsignal(f)| of the fitted model has a low-pass fil-
10−4 10−3 10−2 101 102 103 104 100 101 102 103 ter shape with cutoff at f =10−2 Hz. We predict this shape
f[Hz] T[sec] T[sec]
F G H couldbeprobeddirectlybypeaksinFouriertransformofY
m
10−1 10−1 (‘simulation’),ifsinusoidalinput(Eq. 10)isused. (B) Same
101
AF100 Prob10−3 Prob10−3 mpaordiseolnwthoenexsptiemriumlaetnetdiwnitfrheq1u/efnsctyimruanlagteiosnwphaetrteerhnigihncSoNmR-
allows reliable estimation of S (f).
10−1 10−5 10−6 TY
100 101 102 103 0 10 20 30 0 20 40 60
T[sec] Sequencelength Sequencelength
FIG. 3: The measures of “scale free” rate dynamics in forethescalingbehaviormusthavesomecutoff. Extrap-
the extended stochastic HHS model-comparisonofthe olatingtheexperimentalresultssuggeststhatforα=1.4
experimentaldatafrom[13]andasimulationoftheextended thiscutoffcouldbereachedapproximatelyonlyafterfour
HHS model (solid and dashed lines, respectively). We use
years of ongoing experiments - which is comparable with
here the same measures as in Fig. 6 in [13]: A The firing
the lifespan of a rat (in comparison, α=2 gives a cutoff
rate fluctuations estimated using bins of different sizes (10,
30, 100, 300 sec) and plotted on normalized time axis (units of a few days). In general, M = 5 poles should allow
innumberofbins),aftersubtractingthemeanofeachseries. coverage of all this timescale range [15]. However, in our
B CV of the bin counts, as a function of bin size, plotted on model, the scaling is limited by channel numbers. Since
a log-linear axis. C Firing rate periodogram. D Detrended typically 1 < N < 106 [36], scaling in N can generate
i i
fluctuations analysis. E Fano factor (FF) curve. F Allan f α in the PSD over 6/α Orders of Magnitude (OM).
−
factor (AF) curve. G Length distribution of spike–response
For α = 1.4, we get 4.2 OM, while [13] covered about 3
sequences, on a half-logarithmic axes. H Length distribution
OM.
ofno-spike-responsesequences,onadouble-logarithmicaxes.
For additional details see [13] and references therein. To summarize this section, we used the linearity of
ourderivedI/Otodecomposethecontributionsofinputs
and internal noise to the response of the neuron. This
fact,continuousinputsignalsmayverywellgeneratelong decomposition shows that even though the neuron can
memory effects [33]. “remember” its intrinsic fluctuations over timescales of
days, itsmemoryofpastinputscanberelatively“short”.
Second, we quantify the “noisiness” of the neuronal re-
Forexample,theinputmemoryofourfittedmodeldecay
sponse. In the fitted model the neuronal response under
exponentially with a timescale of 100sec. We suggest
periodicstimulationisverynoisy-inthesensethatlinear
∼
an experiment to test this directly. Additionally, this
optimalestimationofY performssimilarlytothetrivial
m
linear decomposition allows us to quantify the signal-to-
predictor(Y = Y 0.5 ), evenif s isfullyknown
m m m
(cid:100)(cid:104) (cid:105)− (cid:101) noise ratio of the I/O and show that it increases with
(since e has a large variance). In order to improve pre-
m
input spike time variability. Finally, we set upper limits
dictability of Y , we can increase the variability of the
m on the timescale range of the observed 1/fα behavior.
input T . To test this we examined data from a similar
m
experiment where the variability of T was higher than Acknowledgments The authors are grateful to O.
m
theinternalnoiseatcertainfrequencies,soitwaspossible Barak, N. Brenner, Y. Elhanati, A. Gal, T. Knafo, Y.
to estimate S (f)=H (f)S (f) in these regions KafriandS.Maromforinsightfuldiscussionsandreview-
YT signal T
(which is generally hard if internal noise is high [34]), ing parts of this manuscript. The research was partially
which seems to correspond well with our fitted model funded by the Technion V.P.R. fund and by the Intel
(Fig. 4B). Therefore, input variability in inter-spike in- Collaborative Research Institute for Computational In-
tervals T can increase signal to noise ratio, as was pre- telligence (ICRI-CI).
m
viously observed for current amplitude variability over
shorter time scales of milliseconds [35].
Finally, consider the stability of the neuronal response
for even longer timescales. Generally, S (f) f α as
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