Table Of Contentvi
J.A. Guest, Biology and Biochemistry, University of Houston, Houston, TX, USA
C. Hart, Neurobiology and Anatomy, Drexel University College of Medicine, 2900 Queen Lane,
Philadelphia, PA 19129, USA
M.Hawken,CenterforNeuralScience,NewYorkUniversity,4WashingtonPlace,NewYork,NY10003,
USA
M.R.Hinder,PerceptionandMotorSystemsLaboratory,SchoolofHumanMovementStudies,University
of Queensland, Brisbane, Queensland 4072, Australia
G.E. Hinton, Department ofComputer Science, Universityof Toronto, 10Kings College Road, Toronto,
M5S 3G4 Canada
A.Ijspeert,SchoolofComputerandCommunicationSciences,EcolePolytechniqueFe´de´raledeLausanne
(EPFL), Station 14, CH-1015 Lausanne, Switzerland
J.F. Kalaska, GRSNC, De´partement de Physiologie, Faculte´ de Me´decine, Pavillon Paul-G. Desmarais,
Universite´ de Montre´al, C.P. 6128, Succursale Centre-ville, Montre´al, QC H3C 3J7, Canada
M. Kerszberg, Universite´ Pierre et Marie Curie, Mode´lisation Dynamique des Syste`mes Inte´gre´s UMR
CNRS7138—Syste´matique,Adaptation,e´volution,7QuaiSaintBernard,75252ParisCedex05,France
U.Knoblich,CenterforBiologicalandComputationalLearning,McGovernInstituteforBrainResearch,
Computer Science and Artificial Intelligence Laboratory, Brain and Cognitive Sciences Department,
Massachusetts Institute of Technology, 43 Vassar Street #46-5155B, Cambridge, MA 02139, USA
C. Koch, DivisionofBiology, California Instituteof Technology, MC 216-76,Pasadena, CA 91125, USA
J.H. Kotaleski, Computational Biology and Neurocomputing, School of Computer Science and
Communication, Royal Institute of Technology, SE 10044 Stockholm, Sweden
M. Kouh, Center for Biological and Computational Learning, McGovern Institute for Brain Research,
Computer Science and Artificial Intelligence Laboratory, Brain and Cognitive Sciences Department,
Massachusetts Institute of Technology, 43 Vassar Street #46-5155B, Cambridge, MA 02139, USA
A. Kozlov, Computational Biology and Neurocomputing, School of Computer Science and
Communication, Royal Institute of Technology, SE 10044 Stockholm, Sweden
J.W. Krakauer, The Motor Performance Laboratory, Department of Neurology, Columbia University
College of Physicians and Surgeons, New York, NY 10032, USA
G. Kreiman, Department of Ophthalmology and Neuroscience, Children’s Hospital Boston, Harvard
Medical School and Center for Brain Science, Harvard University
N.I.Krouchev,GRSNC, De´partementde Physiologie, Faculte´ de Me´decine, Pavillon Paul-G.Desmarais,
Universite´ de Montre´al, C.P. 6128, Succursale Centre-ville, Montre´al, QC H3C 3J7, Canada
I. Kurtzer, Centre for Neuroscience Studies, Queen’s University, Kingston, ON K7L 3N6, Canada
A. Lansner, Computational Biology and Neurocomputing, School of Computer Science and
Communication, Royal Institute of Technology, SE 10044 Stockholm, Sweden
P.E. Latham, Gatsby Computational Neuroscience Unit, London WC1N 3AR, UK
M.F. Levin, Center for Interdisciplinary Research in Rehabilitation, Rehabilitation Institute of Montreal
and Jewish Rehabilitation Hospital, Laval, QC, Canada
J. Lewi, Georgia Institute of Technology, Atlanta, GA, USA
Y. Liang, Biology and Biochemistry, University of Houston, Houston, TX, USA
W.J. Ma, Department of Brain and Cognitive Sciences, University of Rochester, Rochester, NY 14627,
USA
K.M. MacLeod, Department of Biology, University of Maryland, College Park, MD 20742, USA
L. Maler, Department of Cell and Molecular Medicine and Center for Neural Dynamics, University of
Ottawa, 451 Smyth Rd, Ottawa, ON K1H 8M5, Canada
E. Marder, Volen Center MS 013, Brandeis University, 415 South St., Waltham, MA 02454-9110, USA
S.N. Markin, Department of Neurobiology and Anatomy, Drexel University College of Medicine, 2900
Queen Lane, Philadelphia, PA 19129, USA
vii
N.Y. Masse, Department of Physiology, McGill University, 3655 Sir William Osler, Montreal, QC H3G
1Y6, Canada
D.A. McCrea, Spinal Cord Research Centre and Department of Physiology, University of Manitoba, 730
William Avenue, Winnipeg, MB R3E 3J7, Canada
A.Menciassi,CRIMLaboratory,ScuolaSuperioreSant’Anna,VialeRinaldoPiaggio34,56025Pontedera
(Pisa), Italy
T. Mergner, Neurological University Clinic, Neurocenter, Breisacher Street 64, 79106 Freiburg, Germany
T.E. Milner, School of Kinesiology, Simon Fraser University, Burnaby, BC V5A 1S6, Canada
P. Mohajerian, Computer Science and Neuroscience, University of Southern California, Los Angeles, CA
90089-2905, USA
L.Paninski,DepartmentofStatisticsandCenterforTheoreticalNeuroscience,ColumbiaUniversity,New
York, NY 10027, USA
V. Patil, Neurobiology and Anatomy, Drexel University College of Medicine, 2900 Queen Lane,
Philadelphia, PA 19129, USA
J.F.R. Paton, Department of Physiology, School of Medical Sciences, University of Bristol, Bristol BS8
1TD, UK
J. Pillow, Gatsby Computational Neuroscience Unit, University College London, Alexandra House, 17
Queen Square, London WC1N 3AR, UK
T. Poggio, Center for Biological and Computational Learning, McGovern Institute for Brain Research,
Computer Science and Artificial Intelligence Laboratory, Brain and Cognitive Sciences Department,
Massachusetts Institute of Technology, 43 Vassar Street #46-5155B, Cambridge, MA 02139, USA
A. Pouget, Department of Brain and Cognitive Sciences, University of Rochester, Rochester, NY 14627,
USA
A. Prochazka, Centre for Neuroscience, 507 HMRC University of Alberta, Edmonton, AB T6G 2S2,
Canada
R.Rohrkemper,PhysicsDepartment,InstituteofNeuroinformatics,SwissFederalInstituteofTechnology,
Zu¨rich CH-8057, Switzerland
I.A. Rybak, Department of Neurobiology and Anatomy, Drexel University College of Medicine,
Philadelphia, PA 19129, USA
A. Sangole, Center for Interdisciplinary Research in Rehabilitation, Rehabilitation Institute of Montreal
and Jewish Rehabilitation Hospital, Laval, QC, Canada
S.Schaal,ComputerScienceandNeuroscience,UniversityofSouthernCalifornia,LosAngeles,CA90089-
2905, USA
S.H.Scott,CentreforNeuroscienceStudies,DepartmentofAnatomyandCellBiology,Queen’sUniversity,
Botterell Hall, Kingston, ON K7L 3N6, Canada
T. Serre, Center for Biological and Computational Learning, McGovern Institute for Brain Research,
Computer Science and Artificial Intelligence Laboratory, Brain and Cognitive Sciences Department,
Massachusetts Institute of Technology, 43 Vassar Street #46-5155B, Cambridge, MA 02139, USA
R. Shadmehr, Laboratory for Computational Motor Control, Department of Biomedical Engineering,
Johns Hopkins School of Medicine, Baltimore, MD 21205, USA
R.Shapley,CenterforNeuralScience,NewYorkUniversity,4WashingtonPlace,NewYork,NY10003,
USA
J.C. Smith, Cellular and Systems Neurobiology Section, National Institute of Neurological Disorders and
Stroke, National Institutes of Health, Bethesda, MD 20892-4455, USA
C. Stefanini,CRIM Laboratory,ScuolaSuperiore Sant’Anna,Viale Rinaldo Piaggio34,56025Pontedera
(Pisa), Italy
J.A. Taylor, Department of Biomedical Engineering, Washington University, 1 Brookings Dr., St Louis,
MO 63130, USA
viii
K.A.Thoroughman,DepartmentofBiomedicalEngineering,WashingtonUniversity,1BrookingsDr.,St
Louis, MO 63130, USA
L.H. Ting, The Wallace H. Coulter Department of Biomedical Engineering at Georgia Tech and Emory
University, 313 Ferst Drive, Atlanta, GA 30332-0535, USA
A.-E. Tobin, Volen Center MS 013, Brandeis University, 415 South St., Waltham, MA 02454-9110, USA
E. Torres, Division of Biology, California Institute of Technology, Pasadena, CA 91125, USA
J.Z.Tsien,CenterforSystemsNeurobiology,DepartmentsofPharmacologyandBiomedicalEngineering,
Boston University, Boston, MA 02118, USA
D. Tweed, Departments of Physiology and Medicine, University of Toronto, 1 King’s College Circle,
Toronto, ON M5S 1A8, Canada
D.B. Walther,Beckman InstituteforAdvancedScienceandTechnology, UniversityofIllinoisatUrbana-
Champaign, 405 N. Mathews Ave., Urbana, IL 61801, USA
A.C.Wilhelm,DepartmentofPhysiology,McGillUniversity,3655SirWilliam Osler,Montreal,QCH3G
1Y6, Canada
D. Xing, Center for Neural Science, New York University, 4 Washington Place, New York, NY 10003,
USA
S. Yakovenko, Departe´ment de Physiologie, Universite´ de Montre´al, Pavillon Paul-G. Desmarais,
Universite de Montreal, C.P. 6128, Succ. Centre-ville, Montreal, QC H3C 3J7, Canada
D.Zipser,DepartmentofCognitiveScience,UCSD0515,9500GilmanDrive,SanDiego,CA92093,USA
Preface
In recent years, computational approaches have become an increasingly prominent and influential part of
neuroscienceresearch.Fromthecellularmechanismsofsynaptictransmissionandthegenerationofaction
potentials, to interactions among networks of neurons, to the high-level processes of perception and
memory,computationalmodelsprovidenewsourcesofinsightintothecomplexmachinerywhichunderlies
our behaviour. These models are not merely mathematical surrogates for experimental data. More
importantly, they help us to clarify our understanding of a particular nervous system process or function,
and to guide the design of our experiments by obliging us to express our hypotheses in a language of
mathematicalformalisms.Amathematicalmodelisanexplicithypothesis,inwhichwemustincorporateall
of our beliefs and assumptions in a rigorous and coherent conceptual framework that is subject to
falsification and modification. Furthermore, a successful computational model is a rich source of
predictions for future experiments. Even a simplified computational model can offer insights that unify
phenomena across different levels of analysis, linking cells to networks and networks to behaviour. Over
the last few decades, more and more experimental data have been interpreted from computational
perspectives, new courses and graduate programs have been developed to teach computational
neuroscience methods and a multitude of interdisciplinary conferences and symposia have been organized
to bring mathematical theorists and experimental neuroscientists together.
Thisbookistheresultofonesuchsymposium,heldattheUniversite´ deMontre´alonMay8–9,2006(see:
http://www.grsnc.umontreal.ca/XXVIIIs). It was organized by the Groupe de Recherche sur le Syste`me
Nerveux Central (GRSNC) as one of a series of annual international symposia held on a different topic
eachyear.Thiswasthefirstsymposium inthat annualseries thatfocused oncomputational neuroscience,
and it included presentations by some ofthe pioneers ofcomputational neuroscience aswellas prominent
experimental neuroscientists whose research is increasingly integrated with computational modelling. The
symposium was a resounding success, and it made clear to us that computational models have become a
major and very exciting aspect of neuroscience research. Many of the participants at that meeting have
contributed chapters to this book, including symposium speakers and poster presenters. In addition, we
invited a number of other well-known computational neuroscientists, who could not participate in the
symposium itself, to also submit chapters.
Of course, a collection of 34 chapters cannot cover more than a fraction of the vast range of
computational approaches which exist. We have done our best to include work pertaining to a variety of
neural systems, at many different levels of analysis, from the cellular to the behavioural, from approaches
intimatelytiedwithneuraldatatomoreabstractalgorithmsofmachinelearning.Theresultisacollection
which includes models of signal transduction along dendrites, circuit models of visual processing,
computational analyses of vestibular processing, theories of motor control and learning, machine
algorithms for pattern recognition, as well as many other topics. We asked all of our contributors to
address their chapters to a broad audience of neuroscientists, psychologists, and mathematicians, and to
focus on the broad theoretical issues which tie these fields together.
The conference, and this book, would not have been possible without the generous support of the
GRSNC,theCanadianInstituteofAdvancedResearch(CIAR),InstituteofNeuroscience,MentalHealth
and Addiction (INMHA) of the Canadian Institutes of Health Research (CIHR), the Fonds de la
ix
x
Recherche en Sante´ Que´bec (FRSQ), and the Universite´ de Montre´al. We gratefully acknowledge these
sponsors as well as our contributing authors who dedicated their time to present their perspectives on the
computational principles which underlie our sensations, thoughts, and actions.
Paul Cisek
Trevor Drew
John F. Kalaska
P.Cisek,T.Drew&J.F.Kalaska(Eds.)
ProgressinBrainResearch,Vol.165
ISSN0079-6123
Copyrightr2007ElsevierB.V.Allrightsreserved
CHAPTER 1
The neuronal transfer function: contributions from
voltage- and time-dependent mechanisms
Erik P. Cook1,(cid:1), Aude C. Wilhelm1, Jennifer A. Guest2, Yong Liang2, Nicolas Y. Masse1
and Costa M. Colbert2
1Departmentof Physiology, McGillUniversity, 3655SirWilliam Osler,Montreal, QCH3G 1Y6,Canada
2Biology andBiochemistry, University of Houston,Houston, TX,USA
Abstract: The discoverythatanarray ofvoltage-and time-dependentchannelsispresent inboth the dend-
rites and soma of neurons has led to a variety of models for single-neuron computation. Most of these
models, however, are based on experimental techniques that use simplified inputs of either single synaptic
events or brief current injections. In this study, we used a more complex time-varying input to mimic the
continuous barrage of synaptic input that neurons are likely to receive in vivo. Using dual whole-cell
recordingsofCA1pyramidalneurons,weinjectedlong-durationwhite-noisecurrentintothedendrites.The
amplitudevarianceofthisstimuluswasadjustedtoproduceeitherlowsubthresholdorhighsuprathreshold
fluctuationsofthesomaticmembranepotential.Somaticactionpotentialswereproducedinthehighvariance
inputcondition.Applyingarigoroussystem-identificationapproach,wediscoveredthattheneuronalinput/
output function was extremely well described by a model containing a linear bandpass filter followed by a
nonlinearstatic-gain.Usingcomputermodels,wefoundthatarangeofvoltage-dependentchannelproperties
can readily account for the experimentally observed filtering in the neuronal input/output function. In
addition,thebandpasssignalprocessingoftheneuronalinput/outputfunctionwasdeterminedbythetime-
dependence of the channels. A simple active channel, however, could not account for the experimentally
observed change in gain. These results suggest that nonlinear voltage- and time-dependent channels con-
tributetothelinearfilteringoftheneuronalinput/outputfunctionandthatchannelkineticsshapetemporal
signal processing in dendrites.
Keywords: dendrite; integration; hippocampus; CA1; channel; system-identification; white noise
The neuronal input/output function of understanding information processing in the
brain. The past two decades of research have
What are the rules that single neurons use to significantly increased our knowledge of how neu-
process synaptic input? Put another way, what is ronsintegratesynapticinput,includingthefinding
theneuronalinput/outputfunction?Revealingthe thatdendritescontainnonlinearvoltage-andtime-
answertothisquestioniscentraltothelargertask dependent mechanisms (for review, see Johnston
et al., 1996). However, there is still no consensus
(cid:1) on the precise structure of the rules for synaptic
Correspondingauthor.Tel.:+15143987691;
Fax:+15143988241;E-mail:[email protected] integration.
DOI:10.1016/S0079-6123(06)65001-2 1
2
Early theoretical models of neuronal computa- currentinjection,appliedwithnobackgroundacti-
tion described the neuronal input/output function vity present (but see Larkum et al., 2001; Oviedo
as a static summation of the synaptic inputs and Reyes, 2002; Ulrich, 2002; Oviedo and Reyes,
(McCulloch and Pitts, 1943). Rall later proposed 2005; Gasparini and Magee, 2006). Based on the
that cable theory could account for the passive average spike rate of central neurons, it is unlikely
electrotonic properties of dendritic processing that dendrites receive single synaptic inputs in
(Rall, 1959). This passive theory of dendritic inte- isolation. A more likely scenario is that dendrites
gration has been extremely useful because it receiveconstanttime-varyingexcitatoryandinhibi-
encompassesboththe spatial and temporal aspects torysynapticinput thattogetherproduces random
oftheneuronalinput/outputfunctionusingasingle fluctuations in the membrane potential (Ferster
quantitative framework. For example, the passive and Jagadeesh, 1992; Destexhe and Pare, 1999;
model predicts that the temporal characteristics of Chanceetal.,2002;Destexheetal.,2003;Williams,
dendrites are described by a lowpass filter with 2004). The challenge is to incorporate this type
a cutoff frequency that is inversely related to the of temporally varying input into our study of the
distance from the soma. neuronal input/output function. Fortunately, sys-
Therecentdiscoverythatdendritescontainarich tem-identification theory provides us with several
collection of time- and voltage-dependent channels useful tools for addressing this question.
has renewed and intensified the study of dendritic
signal processing at the electrophysiological level
(forreviews,seeHausseretal.,2000;Magee,2000; Using a white-noise input to reveal the neuronal
SegevandLondon,2000;Reyes,2001;Londonand input/output function
Hausser, 2005). The central goal of this effort has
been to understand how these active mechanisms The field of system-identification theory hasdevel-
augment the passive properties of dendrites. These oped rigorous methods for describing the input/
studies, however, have produced somewhat con- output relationships of unknown systems (for
flicting results as to whether dendrites integrate reviews, see Marmarelis and Marmarelis, 1978;
synaptic inputs in a linear or nonlinear fashion Sakai,1992;WestwickandKearney,2003)andhas
(Urban and Barrionuevo, 1998; Cash and Yuste, been used to describe the relationship between
1999; Nettleton and Spain, 2000; Larkum et al., externalsensoryinputsandneuronalresponsesina
2001;Weietal.,2001;Tamasetal.,2002;Williams varietyofbrainareas(forreviews,seeChichilnisky,
and Stuart, 2002). The focus of past electrophysi- 2001;Wuetal.,2006).Aprominenttoolinsystem-
ological studies has also been to identify the con- identificationistheuseofa‘‘white-noise’’stimulus
ditions in which dendrites initiate action potentials to characterize the system. Such an input theoret-
(Stuart et al., 1997; Golding and Spruston, 1998; icallycontainsalltemporalcorrelationsandpower
Larkum and Zhu, 2002; Ariav et al., 2003; at all frequencies. If the unknown system is linear,
Gasparini et al., 2004; Womack and Khodakhah, orslightlynonlinear,itisastraightforwardprocess
2004), to understand how dendrites spatially and toextractadescriptionofthesystembycorrelating
temporally integrate inputs (Magee, 1999; Polsky the output with the random input stimulus. If the
et al., 2004; Williams, 2004; Gasparini and Magee, unknown systemishighlynonlinear, however,this
2006; Nevian et al., 2007), and to reveal the extent approach is much more difficult.
oflocaldendriticcomputation(Mel,1993;Hausser One difficulty of describing the input/output
and Mel, 2003; Williams and Stuart, 2003). function of a single neuron is that we lack precise
Although these past studies have shed light on statisticaldescriptionsoftheinputsneuronsreceive
many aspects of single-neuron computation, most over time. Given that a typical pyramidal neuron
studieshave focused onquiescentneurons invitro. hasovertenthousandsynapticcontacts,onemight
A common experimental technique is to observe reasonably estimate that an input arrives on the
how dendrites process brief ‘‘single-shock’’ inputs, dendrites every millisecond or less, producing
either a single EPSP or the equivalent dendritic membrane fluctuationsthat are constantly varying
3
in time. Thus, using a white-noise input has two electrodeandmeasuredthemembranepotentialat
advantages: (1) it affords the use of quantitative the soma (V) with a second electrode. The ampli-
s
methods for identifying the dendrite input/output tude distribution of the injected current was
functionand(2)itmayrepresentastimulusthatis Gaussian with zero mean. Electrode separation
statistically closer to the type of input dendrites ranged from 125 to 210mm with the dendrite elec-
receive in vivo. trodeplacedonthemainproximalapicaldendritic
We applied a system-identification approach to branch. Figure 1 illustrates a short segment of
reveal the input/output function of hippocampal the white-noise stimulus and the corresponding
CA1 pyramidal neurons in vitro (Fig. 1). We used somatic membrane potentials.
standard techniques to perform dual whole-cell To examine how the input/output function
patchclamprecordingsinbrainslices(Colbertand changed with different input conditions, we alter-
Pan, 2002). More specifically, we injected 50s of nately changed the variance of the input current
white-noisecurrent(I )intothedendriteswithone between low and high values. The low-variance
d
input produced small subthreshold fluctuations in
the somatic membrane potential. In contrast, the
high-variance input produced large fluctuations
that caused the neurons to fire action potentials
with an average rate of 0.9spikes/s. This rate of
firingwaschosenbecauseitissimilartotheaverage
firing rate of CA1 hippocampal neurons in vivo
(Markus et al., 1995; Yoganarasimha et al., 2006).
Thus, we examined the dendrite-to-soma input/
output function under physiologically reasonable
subthresholdandsuprathresholdoperatingregimes.
The LN model
Figure1illustratesourapproachfordescribingthe
input/output function of the neuron using an LN
model (Hunter and Korenberg, 1986). This is a
functionalmodelthatprovidesanintuitivedescrip-
tion of the system under study and has been parti-
cularly useful for capturing temporal processing in
the retina in response to random visual inputs (for
Fig.1. Usingasystem-identificationapproachtocharacterize
reviews,see Meister and Berry, 1999; Chichilnisky,
the dendrite-to-somainput/outputfunction.(A) Fiftyseconds
2001) and the processing of current injected at
ofzero-meanGaussiandistributedrandomcurrent(I )wasin-
d
jected into the proximal apical dendrites of CA1 pyramidal the soma of neurons (Bryant and Segundo, 1976;
neuronsandthemembranepotential(Vs)wasrecordedatthe Poliakovetal.,1997;Binderetal.,1999;Sleeetal.,
soma. The variance of the injected current was switched be- 2005).TheLNmodelisacascadeoftwoprocessing
tween low (bottom traces) and high (top traces) on alternate
stages:Thefirststageisafilter(the‘‘L’’stage)that
trials.Actionpotentialswereproducedwiththehigh-variance
linearly convolves the input current I . The output
input.(B)AnLNmodelwasfittothesomaticpotential.The d
inputtothemodelwastheinjectedcurrentandtheoutputof of the linear filter, F, is the input to the nonlinear
the model was the predicted soma potential (V^S). The LN second stage(the‘‘N’’stage)thatconvertstheout-
modelwascomposedofalinearfilterthatwasconvolvedwith put of the linear filter into the predicted somatic
theinputcurrentfollowedbyastatic-gainfunction.Theoutput potentials (V^ ). This second stage is static and can
ofthelinearfilter,F(arbitraryunits),wasscaledbythestatic- S
be viewed as capturing the gain of the system. The
gainfunctiontoproducethepredictedsomaticpotential.The
static-gainfunctionwasmodeledasaquadraticfunctionofF. two stages of the LN model are represented
4
mathematically as input/output function is shown in Fig. 2. For
this neuron, the LN model’s predicted somatic
F ¼H(cid:1)I
d membrane voltage (V^ ; dashed line) almost
S
V^S ¼GðFÞ ð1Þ perfectly overlapped the neuron’s actual somatic
potential (V, thick gray line) for both input condi-
where H is a linear filter, * the convolution oper- s
tions (Fig. 2A and B). The LN model was able to
ator, and G a quadratic static-gain function.
fully describe the somatic potentials in response to
Havingtwostagesofprocessingisanimportant
the random input current with very little error.
aspectofthemodelbecauseitallowsustoseparate
Computing the Pearson’s correlation coefficient
temporal processing from gain control. The linear
over the entire 50s of data, the LN model
filter describes the temporal processing while the
accounted for greater than 97% of the variance of
nonlinear static-gain captures amplitude-depend-
this neuron’s somatic potential.
ent changes in gain. Thus, this functional model
Repeating this experiment in 11 CA1 neurons,
permits us to describe the neuronal input/output
the LN model accounted for practically all of the
functionusingquantitativelyprecisetermssuchas
somatic membrane potential (average R240.97).
filtering and gain control. In contrast, highly de-
Both the low and high variance input conditions
tailed biophysical models of single neurons, with
werecapturedequallywellbytheLNmodel.Thus,
their large number of nonlinear free parameters,
the LN model is a functional model that describes
are less likely to provide such a functionally clear
theneuronalinput/outputfunctionoverarangeof
description of single-neuron computation.
input regimes from low-variance subthreshold to
It is important to note that we did not seek to
high-variance suprathreshold stimulation.
describe the production of action potentials in the
dendrite-to-soma input/output function. Action
potentialsareextremelynonlineareventsandwould
Gain but not filtering adapts to the input variance
notbecapturedbytheLNmodel.Weinsteadfocu-
sed on explaining the subthreshold fluctuations of
The LN model’s linear filters and nonlinear static-
the somatic voltage potential. Thus, action poten-
gainfunctionsareshownforourexampleneuronin
tials were removed from the somatic potential be-
Fig. 2C and D. The impulse-response function of
forethedatawereanalyzed.Thiswasaccomplished
the linear filters (Fig. 2C) for both the low (solid
bylinearlyinterpolatingthesomaticpotentialfrom
line) and high (dashed line) variance inputs had
1msbeforetheoccurrenceoftheactionpotentialto
pronounced negativities corresponding to a band-
either5or10msaftertheactionpotential.Because
pass in the 1–10Hz frequency range (inset).
action potentials make up a very small part of the
Although the two input conditions were signifi-
50s of data (typically less than 2%), our results
cantly different, the filters for the low- and high-
werenotqualitativelyaffectedwhenthespikeswere
variance inputs were very similar. Across our pop-
left in place during the analysis.
ulationofneurons,wefoundnosystematicchange
inthelinearfiltersastheinputvariancewasvaried
The LN model accounts for the dendrite-to-soma between low and high levels. Therefore, the tem-
input/output function poral processing performed by CA1 pyramidal
neurons on inputs arriving at the proximal apical
Usingstandard techniques,wefittheLNmodelto dendrites does not change with the input variance.
reproduce the recorded somatic potential in res- IncontrasttothefilteringpropertiesofCA1neu-
ponsetotheinjecteddendriticcurrent(Hunterand rons, the static-gain function changed as a function
Korenberg,1986).Wewantedtoknowhowthelow of input variance. Figure 2D illustrates the static-
and high variance input conditions affected the gain function for both input conditions. In this
components of the LN model. Therefore, these plot,therestingmembranepotentialcorrespondsto
conditions were fit separately. An example of 0mVandtheunitsfortheoutputofthelinearfilter
the LN model’s ability to account for the neuronal (F) are arbitrary. The static-gain function for the
5
Fig.2. Thedendrite-to-somainput/outputfunctionofaCA1neuroniswelldescribedbytheLNmodel.(A)Exampleof500msofthe
input current and somatic potential for the low-variance input. The predicted somatic membrane potential of the LN model (V^ ;
S
dashedline)overlapstherecordedsomaticpotential(V,thickgrayline).(B)ExampleoftheLNmodel’sfittothehigh-varianceinput.
s
ActionpotentialswereremovedfromtherecordedsomaticpotentialbeforefittingtheLNmodeltothedata.(C)Theimpulse-response
functionofthelinearfiltersfortheoptimizedLNmodelcorrespondingtothelow(solidline)andhigh(dashedline)varianceinputs.
Insetisthefrequencyresponseofthefilters.(D)Static-gainfunctionfortheoptimizedLNmodelplottedforthelow(solidline)and
high(dashedline)varianceinputs.Theaxesforthehighvarianceinputwereappropriatelyscaledsothattheslopeofbothstatic-gain
functionscouldbecompared.
low-varianceinputwasastraightlineindicatingthat the resting membrane potential. This reduction in
the neuronal input/output function was linear. For gain also increased with both hyperpolarized and
the high-variance input, however, the static-gain depolarizedpotentials.Adaptingtothevarianceof
function demonstrated two important nonlineari- an input is an important form of gain control
ties. First, the static-gain function showed a com- because it ensures that the input stays within the
pressive nonlinearity at depolarized potentials. operating range of the neuron. Although a 16%
Thus, at large depolarizing potentials, there was a reduction may seem small in comparison to the
reduction in the gain of the input/output relation- largechangeintheinput-variance,therearemany
ship.Second,therewasageneralreductioninslope instanceswheresmallchanges inneuronalactivity
of the static-gain function for high-variance input are related to significant changes in behavior. For
comparedwiththelow-varianceslope,indicatingan visual cortical neurons, it has been shown that
overall reduction in gain. Thus, for this neuron, in- small changes in spike activity (o5%) are corre-
creasing the variance of the input reduced the gain lated with pronounced changes in perceptual
oftheinput/outputfunctionatrestthatwasfurther abilities (Britten et al., 1996; Dodd et al., 2001;
reduced for depolarizing potentials. Cook and Maunsell, 2002; Uka and DeAngelis,
Across our population of 11 neurons, we found 2004; Purushothaman and Bradley, 2005). Thus,
that increasing the variance of the input reduced even small modulations of neuronal activity can
the gain of CA1 neurons by an average of 16% at have large effects on behavior.
