Analysis and modeling of low frequency local field oscillations in a hippocampus circuit under osmotic challenge: the possible role of arginine vasopressin circuit for hippocampal function
Hernan Barrio Zhang, Mariana Marquez-Machorro, Vito S. Hernandez, Andres Molina, Limei Zhang, Tzipe Govezensky, Rafael A. Barrio
AAnalysis and modeling of low frequency local field oscillations in a hippocampuscircuit under osmotic challenge: the possible role of arginine vasopressin circuit forhippocampal function.
Hern´an Barrio Zhang , Mariana M´arquez-Machorro , Vito S. Hern´andez ,Andr´es Molina , Limei Zhang , Tzipe Govezensky , and Rafael A. Barrio ∗ Facultad de Ciencias, U.N.A.M., 01000 - M´exico D.F., Mexico. Facultad de Medicina, Departamento de Fisiolog´ıa,Universidad Nacional Aut´onoma de M´exico, Mexico. Instituto de Investigaciones Biom´edicas, Universidad Nacional Aut´onoma de M´exico, 04510 M´exico D.F., Mexico, Instituto de F´ısica, U.N.A.M., Circuito de la Investigaci´on Cient´ıficaCiudad Universitaria, C.P. 04510, M´exico D.F., Mexico, and McGill University, Canada. (Dated: November 10, 2020)Electrophysiological time series were taken simultaneously in two locations in the hippocampusof a rat brain previously described as receiving innervation from the osmosensitive vasopressinergicneurons of the hypothalamus. A hyperosmotic saline solution injection was administered during thetime of the experiment. We analyze the recorded time series using different methods. We detect amodification of the delta and theta oscillations just after the perturbation caused by the injection.We compare the quality and information that each one of the methods exhibit and we analyzethe characteristics of the perturbation based on a hypothesis that the strength of the functionalconnections between the vasopressinergic hypothalamic magnocellular neurons and their target inthe hippocampus is modified by the perturbation. We built a model of the hypothetic neuralconnections and numerically calculate the time series produced by the system when simulatingthe perturbation caused by the saline injection. The theoretical results resemble the experimentalfindings concerning the frequency and amplitude alterations of the delta and theta bands.
I. INTRODUCTION
The study and analysis of time series of recorded datais a powerful tool to investigate the dynamics and in-teractions of physical, chemical or biological processes inmany and diverse systems. This is particularly true in thestudy of the brain functions, since the brain activity canbe measured by the collective electrical activity of neu-rons and their activation or inhibition by their synapticconnexions and other means of communication.This electrical activity is usually detected by en-cephalograms, MRI, or more precise electrophysiologicalrecordings. The time series collected this way containmuch information about particular time-space correla-tions of the electrical activity of the brain. It is thereforeimportant to use the appropriate methods of analyzingthese data in order to maximize the amount of informa-tion that could be extracted from them.The interactions between different parts of the brainare regulated by many substances called neuromodula-tors that certainly modify the transfer of signals fromone neuron to the other. An important neuromodula-tor is vasopressin, also called the antidiuretic hormone,that is known to control water homeostasis and bloodpressure. It is released by the pituitary into the bloodstream, but also directly to the brain, where it is believedto play an important role in social behavior, sexual con- ∗ Electronic address: barrio@fisica.unam.mx duct, motivation, learning and memory, and response tostress.In this work we record electrophysiological time seriessimultaneously from two specific parts of the brain thatare known to contain vasopressinergic innervation fromthe hypothalamus and that could be modified by per-turbing the hydroelectrolitic homeostasis. More specif-ically, we want to study the time varying properties ofnon stationary changes in the local field potentials andthe coherence in the dorsal and ventral regions of the hip-pocampus of a rat. The signals are taken simultaneouslyin both regions and the rat subjected to the activation ofthe hypothalamic vasopressinergic system by a systemicinjection of a hyperosmotic saline solution.We use different approaches to analyze the data andcompare the information extracted from each one ofthem. We also develop a theoretical model based on theassumption that the external perturbation modifies thecommunication pathways between the regulating systemlocalized in the hypothalamus and the hippocampal re-gions.
II. BIOLOGICAL FACTS
Central nervous system neurons exchange informationvia electrical currents, the sum of all the electrical cur-rents generates changes in the electrical potential of theextracellular medium that vary in time. The low fre-quency ( <
500 Hz) component of these signals is knownas the local field potential (LFP) and can be recorded and a r X i v : . [ q - b i o . N C ] N ov studied from a local network of neurons. Fluctuations inthe amplitude of the LFP can be measured in differentparts of the brain [1], and recent findings indicate thatnetwork oscillations bias input selection, temporally linkneurons into assemblies, and facilitate synaptic plasticity,mechanisms that cooperatively support temporal repre-sentation and long-term consolidation of information [2].Using power spectra, LFP oscillations have been di-vided into several arbitrarily defined frequency bands,including the internationally agreed delta (0.1 - 4 Hz),theta (4-8 Hz), alpha (8-15 Hz), beta (15-30 Hz) andgamma (30 -80 Hz) bands [3]. The presence and powerof these frequency bands have been associated with dif-ferent mental states, tasks and behaviors [4]. However,there are some limitations in this classification since somephysiologically relevant rhythms sometimes fall into twocategories, for example in the awake rodent, hippocampaltheta oscillations fall between 4 and 10 Hz [5].The hippocampus is one of the most studied regionsof the brain, one of its characteristics is the presenceof prominent theta oscillations related to: REM sleep,spatial representations, attention, arousal or anxiety [6].The hippocampus can be divided in functionally differ-ent, albeit connected, dorsal (dHi) and a ventral (vHi)subregions mediating learning/memory and emotionalcontrol/stress responsivity functions, respectively [7].Some recent studies suggest that theta oscillation co-herence between these subregions may increase duringstressful situations [8].We have previously demonstrated that magnocellularneurons located in the supraoptic and paraventricular nu-clei of the hypothalamus , known by their key role in thehomeostatic control of hydroelectrolite balance, in addi-tion to their peripheral axonal projections, send ascend-ing projections to intracerebral limbic targets includingthe dorsal (dHi) and ventral (vHi) hippocampus [9–11](see Fig. 1), this finding suggest that the hypothalamuscould modulate the activity of local neuronal networks inthe dHi and in the vHI, modifying the oscillatory activityand the coupling between the dHi and vHI and thus havean influence in the integration of the cognitive-emotionalfunctions of these structures. III. METHODSA. Animals
Experiments were performed on adult male WistarRats (280-300g), provided by the local animal facility andhoused at 20 − ◦ C on a 12h dark/light cycle (lights onat 19:00h) with tap water and standard rat chow pel-lets available ad libitum. All surgical and experimentalprocedures were performed in accordance with the guide-lines published in the National Institutes of Health Guidefor the Care and Use of Laboratory Animals (publica-tion number 86-23, revised 1987) and with the approvalof the local bioethical and research committee (CIEFM-
FIG. 1: The dorsal (dHi) and ventral (vHi) hippocampusare innervated by vasopressinergic fibers originated in the hy-pothalamic paraventricular (PVN) nuclei. A and A1 Low andhigh power microphotograph of dorsal hippocampus showingfibers (arrows) immunolabelled with an antibody against va-sopressin. B and B1: Low and high power microphotograph ofventral hippocampus showing fibers (arrows) immunolabelledas mentioned. C: Schematic representation of the vasopressin-ergic pathways (orange arrows) by which vasopressin fibersoriginating in the magnocellular neurons of the PVN (Redovals) reach the dorsal (dHi) and ventral (vHi) hippocampi,the arrow thickness represent the density of fibers in each ofthese pathways. The sites of LFP recordings in the dorsaland ventral hippocampus are indicated.
B. Electrophysiological recordings
For in vivo extracellular recording, rats were in-duced into anesthesia with 4% isourane in oxygen, fol-lowed by urethane injection (intraperitoneal, 1.3 g/kg,Sigma-Aldrich), with supplemental doses of xylazine (30mg/kg), as necessary. Body temperature was maintainedat 36 ◦ C with a heating pad.Once anesthetized, animals were placed on a stereo-taxic frame and two craniotomies were performed inorder to position the tip of two glass electrodes (8-15MOhms) coated with DiI (a lipophilic fluorescent dyethat allowed to confirm the correct positioning of theelectrodes) at previously standardized coordinates[12] indHi and vHi, selected based on our previous work defin-ing the main sites of vasopressinergic innervation fromthe hypothalamic vasopressinergic neurons: CA1/CA2region of dorsal hippocampus (coordinates : -2.3 mmposterior to Bregma, 2 mm lateral to midline and 3mm ventral to skull surface) and CA1 ventral hippocam-pus region (ipsilateral coordinates: -4.8 mm posterior toBregma, 5 mm lateral to midline and 8.5 mm ventralto skull surface). Glass monopolar recording electrodeswere referenced against a wire implanted subcutaneouslyin the neck. Extracellular LFP signals from both elec-trodes were amplified (ELC-01MX amplifier, npi elec-tronics, GmbH, Tamm, Germany), filtered between 0.3and 500 Hz (BF-48DGX Filter, npi electronics), digi-tized at 1000 Hz (INT-20X breakout box module con-nected with a National Instruments, NI-M series board)and displayed with the Sciworks Experimenter software(Datawave technologies)
C. Osmotic Stimulation
To activate the hypothalamic vasopressinergic system,after at least 20 minutes of stable basal LFP recordinghad elapsed, a 2% body weight volume of a 900 mM NaClsolution was intraperitoneally injected. Data recordingcontinued during at least 45 minutes after the NaCl in-jection.We performed numerous experiments following theabove mentioned conditions. For illustration purposeswe chose a representative set of data.
D. Histological examination
After the experiment, the rats were given a lethal doseof sodium pentobarbital (63 mg/kg, Sedalpharma, Mex-ico), and transcardially perfused with isotonic NaCl fol-lowed by paraformaldehyde fixative (4% paraformalde-hyde, 15% v/v picric acid in 0.1M phosphate buffer), thebrains were collected, coronally sectioned at 70 µ m in avibratome (Leica VT-1000s). Sections were mounted inmicroscope slides and were observed under fluorescencemicroscopy to verify the correct positioning of the elec-trodes in dHi and vHi. IV. TIME SERIES ANALYSIS
In this section we explain the methods used to analyzethe experimental data. For the sake of clarity, we showresults extracted from a single set of representative Data. These were taken every millisecond thus each record con-sists of 3 million number pairs. Fig.1E is a representativediagram to show the position of the electrodes recordingdata of series v1( t ) (dHi) and v2( t ) (vHi).In order to see the distribution of frequencies in thedata the first thing to do is to calculate the Fourier powerspectra of the time series, which is defined as,Power v i ( ω ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) t f t i v i ( t ) e iwt dt (cid:12)(cid:12)(cid:12)(cid:12) (1)where the integral is taken from the initial time t i to thefinal time t f . FIG. 2: Fourier power spectra and corresponding spectro-grams of the time series v1 and v2.
In the left column of Fig. 2 we show the power spectraof the filtered data. Notice that in Power dHi there is abroad bands with a conspicuous peak at ∼ n Fourier power spectra takenfrom a small time window w ( j ) = [ t i ( j ) , t f ( j )].Then, the frequency dependent power spectra are plot-ted in the y − z plane, and the successive n spectra arestacked along the x axis, which then contains the timewindows as they are sliding along time (notice that thewindows could overlap or not). The three dimensionalplot is usually showing the x − y plane and the z planeis represented with a color code. In the right column ofFig. 2 we show the spectrograms of the two time seriesusing the Fourier spectra of the data. In all spectrogramsshown here we used a window of w ( j ) = 4096 s, with anoverlap of 2048 s.A better way to investigate repeating patterns in thedata is to calculate the self correlation of the series, de-fined as R ff ( τ ) = (cid:90) ∞−∞ f ( t ) f ∗ ( t − τ ) dt = (cid:90) ∞−∞ f ( t + τ ) f ∗ ( t ) dt. (2)and then calculate the power spectra of the self correla-tion. The self correlation emphasizes the real oscillationsover the background. In the left column of Fig. 3 weshow the calculation of the self correlation power spec-tra for the two series. Notice that now the frequencybands are better defined, with a good separation betweenbands, now the bands at 2 Hz and 6Hz (only hinted inthe Fourier spectrum of dHi) are clearly visible, and aband at very low frequency appears in vHi. FIG. 3: Power spectra and spectrograms of the self correla-tions of the same data as in Fig. 2.
In the right column of Fig. 3 we notice that the spectro-gram obtained with the self correlation spectra is muchclearer than than the one obtained with the original data.In both we notice that the particular external event at t p = 1000 seconds (the injection) caused a substantialchange in the bands.The low frequency band ( ∼ ∼ t p = 1000s and t r = 1300 s. The interval from t p to t r should beconsidered as the time the system takes to reach a newstable state after the injection.The cross correlation of the two time series will de-tect correlated signals, it detects periods of simultaneous behavior, or when there is a delay between the same be-havior in both signals. This quantity is defined as C v1 , v2 = (cid:90) ∞−∞ v1( t )v2 ∗ ( t + τ ) dt. (3) FIG. 4: (A) Cross correlation function of the filtered data.In the inset we show a small interval of time around zero, toillustrate the asymmetry of the cross correlation. (B) Thecorresponding power spectrum.
In the upper panel of Fig. 4 we show the cross corre-lation function C dHi , vHi . Notice that the plot is almostsymmetric for positive and negative time differences. Thegreatest correlation is apparently at zero, indicating thatchanges were almost simultaneous. However, in the in-set we show and amplification around zero to verify thatthere is a small delay of the cross correlation of the or-der of some milliseconds. In the lower panel we show thepower spectrum of the cross correlation function. Thethree low frequency bands are clearly noticeable.Our conclusion is that the power spectrum of the crosscorrelation is a better tool to investigate the main recur-rent frequencies present in both series, since neurophysio-logical signals usually repeat in different places with sometime delays, which seems to be the case here. In orderto acquire time information, as well as synchronous ordelayed frequencies, one can construct the spectrogramwith the cross correlations instead of the self correlations.In Fig. 5 we show such a calculation. Notice that the in-formation given by the self correlation spectrograms isseen more clearly in this figure, including very small dis-turbances in the ∼ FIG. 5: Spectrogram using the cross correlation function ofthe filtered data. The color scale is in this case linear.
The final method used is to examine scalograms ob-tained with wavelets. The continuous wavelet transform,unlike the Fourier transform, is able to represent the datain a time-frequency domain very precisely, and it does notrequire the assumption of stationarity. It is defined as ausual convolution, W v ( φ, a, b ) = 1 √ a (cid:90) ∞−∞ v( t ) φ ∗ (cid:18) t − ba (cid:19) dt, (4)where φ is a continuous function of time called the motherwavelet, a is a scale, and b is a time translation.In this work we have used the so called Morlet motherwavelet, whose Fourier transform is defined as φ ( sω ) = 1 π / e − ( sω − ω ) / Θ( sω ) , (5)where Θ is the step (Heaviside) function, s is a scale thatshould be larger than twice the sampling period, and ω isthe mean frequency. In our case we chose zero mean. Animportant issue in wavelet analysis is the correct choice ofscales, which depends on the frequency range one wantsto be focusing and on the time resolution one desires.In Fig. 6 we show the results for the same data, so onecould compare this scalogram with the spectrograms ob-tained with the former methods. One notices the coher-ence of the bands during the perturbation. The waveletcorrelations C (cid:48) v1 , v2 are the equivalent of Eq. 3, but us-ing the wavelets W instead of the bare signals, In thetop two panels of the figure we show C (cid:48) vi , v1 and C (cid:48) v2 , v2 ,respectively and in the lower panel we show the waveletcoherence, defined as ( C (cid:48) v1 , v2 ) / ( C (cid:48) v1 , v1 C (cid:48) v2 , v2 ). Note alsothe continuous coherence of all bands at all times.Since the wavelet transforms are complex, the waveletcross correlation ( C (cid:48) v1 , v2 ) contains information about themodulus and the phase. In Fig. 7 we plot the modulus (cid:113) [ (cid:60) ( C (cid:48) v1 , v2 )] + [ (cid:61) ( C (cid:48) v1 , v2 ) ] on the top and the phaseangle arctan[ (cid:60) ( C (cid:48) v1 , v2 ) / (cid:61) ( C (cid:48) v1 , v2 )] at the bottom.Observe that the modulus shows an enhancement ofthe band at 4 Hz during the perturbation, and that the FIG. 6: Scalograms (spectrograms for wavelets) of the samedata using wavelet analysis phase is locked at zero in all bands also in this period.After recovery from the perturbation the phase of thebands at 2 and 6 Hz are locked at values near ± π .In particular, the band at 6 Hz, which is initially notin phase, is gradually locked at 180 ◦ after the challenge[17]. The delay observing this increase in the phase lock-ing, probably reflects the time necessary for the centralvasopressin system to react and reach a new stable stateafter a hypertonic challenge, in accordance with the re-sults of Ref. [14], where it is shown that the vasopressinconcentration in plasma after a systemic hyperosmoticchallenge increase reaches a maximum 30 minutes afterthe hypertonic challenge.In Fig. 8 we show a comparison of the methods usedfor another Data set. The color code has been changedto a scale from dark blue to light yellow for the sake ofclarity.These results suggest that the prolonged activation ofthe vasopressinergic system can modulate theta activityin the hippocampus and promote the synchronization be-tween the dorsal and ventral regions. V. THEORETICAL MODEL
The experimental results can be understood if one as-sumes that there are two physiologically different butanatomically connected systems in the hippocampus, andthat the neurons in these systems are sensitive to the ef-fect of vasopressin.
FIG. 7: Scalograms from the modulus and the phase of thewavelet cross correlation of the signals. The colour code hasbeen changed to a scale from dark blue to light yellow.FIG. 8: Spectrograms based on Fourier (left column), Corre-lation (central column) and wavelet analysis (right column).In the left bottom corner we also show the wavelet coherenceplot.
One could imagine that the functional connectivity be-tween these two systems is dependent on the localiza-tion/nature of the responsive neurons and on the inner-vation density/local concentrations of vasopressin in thehippocampus. Therefore, if one produces an external sig-nal that potentiates the release from axon terminals ofosmosensitive neurons located in the hypothalamus , thestrength of the connections between the dorsal and ven-tral hippocampus systems change. In order to verify thishypothesis, one could build a simple model in which in-dividual neurons are represented as a set of non lineardynamical equations of the Hodgkin-Huxley type. Wepropose the following model for an inhibitory neuron [15],which is general enough to serve our purposes, C m ∂V∂t = − ( I c + I L + I s ) (6)where C m is the capacitance and V is the membranepotential. There are three currents: the calcium ion cur-rent, I c = g c mh ( V − V c ) , a leak current I L = g L ( V − V L ) , and a synaptic current I s = g s s ( V − V s ) . This model could describe a network of neurons con-nected by synapses of the GABA type. The postsynapticinteraction strength in this model is taken into accountby the dynamic variable s = (cid:80) i s i /N , which is the av-erage of all the i connections between the neuron withall the N members of the network. The kinetic variablesobey the following dynamic equations, ∂s∂t = k f F ( V )(1 − s ) − s/t s ∂h∂t = ( h ∞ ( V ) − h ) t h ( V ) m = m ∞ ( V ) (7)The asymptotic values of the kinetic variables are m ∞ ( V ) = 11 + e − ( V +40) / . ,h ∞ ( V ) = 11 + e ( V +70) / , the h time scale is t h ( V ) /φ = t + t / [1 + e ( V +50) / ] , where t = 30 and t = 500 are times expressed in unitsof φ which could be adjusted to get their value in mil-liseconds. The presynaptic depolarization is chosen as F ∞ ( V ) = 11 + e − ( V +35) / , so values higher than -35 mV will open the synaptic chan-nels.The synapsis decay time t s must be larger than a cer-tain critical value in order to have oscillations. The nu-merical values for the physiological parameters that pro-duce a rhythmic firing pattern are C m = 1( µF cm − ), g L = 0 . g c = 3 . g s = 2, (in mili Siemens cm − ), t s = 16 (in ms ), φ = 0 . k f = 0 . ms − ).The reversal potentials in mV are V L = − V c = 90y V s = − dt = 0 .
1. The actual interspike time interval can beadjusted to the experimental data by changing the timescale units of the time step properly.
FIG. 9: From left to right: Membrane Potential of a singleneuron as a function of time, Power Spectra and Phase Por-trait in the { h, V, s } space, for the parameter values in thetext and : (a) g c = 1 .
23, (b) g c = 2 .
50, (c) g c = 3 . In Fig. 9 we show the time series of the membrane po-tential calculated numerically with this model. Observethat one could change the firing frequency by varying theparameters of the model. In this case we show the vari-ations when using different values of g c . The calculationis for a single neuron with an initial value of the synap-tic variable s = 0 . h, V, s ) space.Now, we shall use this model to built a network thatrepresents the hippocampus. Given the anatomy of thissystem, a good model for the oscillations in it would be alinear chain of a number of neuron models connected bysynaptic interactions, which should be random, in orderto account for alterations in the local field by signalsproduced in the three dimensional vicinity.In Fig. 10 we represent these single neuron models asblue circles and their noisy connections with black lines.Following our basic hypothesis, we need a pace-makersystem that fires periodically. This system is representedby the green circle and it is connected to two locations inthe chain by interactions of different strength (red andyellow lines in the figure). The strength of these con-nections may vary when an external event produces aperturbation in the system. This perturbation in the lo-cal fields could be measured at different locations simul- … FIG. 10: Diagram of the model. Each circle represent a mea-suring location in the hippocampus, all connected by blacknoisy lines. Neuron 1=N is a signaling neuron in the hy-pothalamus that is directed to the hippocampus in differentlocations. The strength of the black, red and yellow connec-tions changes with the perturbation. taneously, in our case, in the experiment one measurestwo time series of the local fields in different locationsof the CA1/CA2 hippocampal layer. In our model theselocations are the blue circles 2 and N-1.One now has a system of coupled non linear noisy os-cillators, which means that the quantities V , h , and s arevectors with N entries.The initial conditions are V inic = − . s inic = . h inic = 0 . , for all N systems.One defines a specific model by setting the strength ofthe synaptic interactions between pairs of systems s ( i,j ) .For systems in the chain ( i = 2 , , ..N − , N −
1) we define s ( i ) = αs ( i − ,i ) + βs ( i +1 ,i ) . On the cells representing theother regulatory system one has s (1) = α s (1 , + β s (2 , and s ( N ) = α N s ( N,N ) + β N s ( N − ,N ) . The strength of the α, β interactions could vary due to the external pertur-bation at a certain time t r − t p .Additionally one should introduce noise in all threedynamical variables to account for the action of the in-teractions with other cells not considered in the model. VI. RESULTS
In order to show the sort of results that one obtainswith this model, we performed a calculation starting withthe synaptic interactions defined above and perturbingthe system at certain times. Our goal is to verify if areasonable hypothesis related with the experimental sit-uation could give similar results.We assume that the experimental situation is such thatthe regions vH and dH are weakly and bidirectionallycommunicating with each other. A third region in thehypothalamus (PVN) communicates with the two for-mer regions, with a strength PVN-vH (cid:29)
PVN-dh (Onedetects many more axons towards vH than to dH). Theseconnections should be modified by the perturbing injec-tion.We chose a chain with N = 50 and integrated thedynamical system for N I = 2 time steps of length dt =0 . N and they allow to simulatethe system for a time lapse comparable to the actualrecordings.We start by adjusting the frequency bands obtainedwith the model to the ones observed in the unperturbedsystem. A stability analysis performed in the ( α i , β i )space reveals that the interactions along the chain shouldbe asymmetric, that is, α (cid:54) = β and that the action of theexternal cell is also asymmetric.Therefore, we start the calculation with the initial con-ditions α = 0, β = 1 . α = 1 . β = 0 . α N = 1 . β = 0 . α = 1 . β = 0, α = 1 . β = 0 . α N = 0 .
2, and β N = 0 . t p = 900 s and t r = 1300 s. Then we restorethe system to the original state at time t r + dt . We alsointroduce white noise of the order of five percent in alldynamical variables, in order to account for local fieldvariations not taken into account in the chain model.In Fig. 11 we show the spectra obtained with the corre-lations. Notice that we catch the main bands appearingin the experimental data shown, with the correct ampli-tude. The spectrum of the external neuron presents abroad band between 10 and 15 Hz. FIG. 11: Power spectra of the self correlation of site 2 ( C ),site N-1 ( C N − ), the cross correlation and in the master neu-ron ( C ). The origin of these bands can be investigated by look-ing at the spectrograms of the analysis of the theoreticaltime series. In Fig. 12 we show these spectrograms andcompare them with the corresponding ones obtained fromthe experimental Data shown before.In panel (A) we show the Fourier analysis, including (A)(B) (C)
FIG. 12: (A) Comparison of the experimental Fourier anal-ysis (on the left) and from the theoretical model on the right.(B) Same comparison for the correlation analysis and (C) forthe wavelet spectrograms. the spectral bands. Notice the differences with the spec-tra obtained with correlations. The comparison with thecorrelation spectrograms is not good because the datacontain four external signals not related with the exper-iment in different times. It is worth noticing that thewavelet correlation of the delta bands almost disappearsduring the perturbation and that the wavelet coherenceis high everywhere and particularly during the perturba-tion. The main band at 4 Hz is present only on site v1( C ) and goes to lower frequency during the perturba-tion, as in the experiment. The asymmetry of all bandsis also comparable to the experimental data. (A) (C)(B) FIG. 13: Same comparison of experimental and model resultsas in Fig. 12
One could use the results of the model to test if agiven hypothesis of the processes that occur in the ex-periment during a controlled perturbation produces theeffects shown in the analysis of the experimental data.One could try a different working hypothesis to in-terpret the data. For instance, if one proposes thatthe interactions are symmetric along the chain, that is s ( i ) = ( s i − ,i + s i +1 ,i ), and that the external signals arequite small α = α N − = β = β N − = 0 .
2. One thenassumes that the effect of the perturbation only consistsof increasing the external signal, or α = α N − = β = β N − = 1.Then one performs the calculations with the samemodel, but in this case one obtains very different results,which have very little to do with the experimental find-ings. This is shown in Fig. 13. Notice that the modelpredicts very different results, which means that it couldbe used to help designing new experiments to validatethe hypothesis. VII. DISCUSSION AND CONCLUSIONS
We shall start by discussing the methods used to ex-amine the experimental data. Fourier power spectra hadbeen successfully used as a first step to detect the mainfrequencies present in time series data. Fourier transformis a summation over the whole recording time because itassumes stationarity; it does not preserve any informa-tion about time. As a consequence, frequencies presentfor short periods may not appear in the spectrum, or theiramplitude may be very small. In the example shown onthe left hand side column of Fig. 2 we notice that theprincipal band in dHi is at 4 Hz while in vHi there aretwo clear bands at 2 and 6 Hz. These bands are notpresent all the time. The Fourier windowed transformfor creating spectrograms were proposed for analyzingseries which change in time. In the spectrograms shownin Fig. 2 we find that the band at 2Hz appears in the dHiduring the perturbation but it disappears afterwards. theFourier spectrum for vHi does show a ∼ b , time information is kept,while by modifying the scale a different frequencies aredetected. Wavelet coherence allows the identification ofsynchronous signals. In Fig. 6 one clearly detects andenhancement of the wavelet correlation between 900 and1300 s, meaning that the perturbation has caused a syn-chronization of both signals in the frequency range ofinterest.Using complex wavelets additionally preserves infor-mation about the phase at every time measured, in ourexample we observe a phase locking of all frequencies atzero angle (meaning that signals are synchronized) duringthe perturbation, confirming the observations of waveletcoherence.The hippocampus is a major part of the limbic systemthat participates in the control of many physiological andbehavioral processes. Theta rhythm is the most promi-nent activity in the hippocampus and is suggested to beinvolved in the flow of information between hippocampalregions. With the aid of the previous methods for analy-sis of time series, the effect of an hyperosmotic stimulithat increase the activity of hypothalamic vasopressinneurons, that we have recently shown to innervate thedorsal and the ventral hippocampus can be evaluated, inparticular the relationship between theta (4-10 Hz) oscil-lations of dHi and vHi can be quantified.The spectrogram of the cross correlation (Fig. 5) showsthat at time 1300s (5 minutes after the stimulus, the timenecessary to reach a new equilibrium) the band at 6 Hzthat initially decreased its frequency gradually increasesuntil the end of the recording reaching a slightly higherfrequency. It is also evident in this figure that after thehypertonic stimulus a `Oband ´O around 4 Hz appears, de-noting increased correlation between the two time series.Using the more temporally sensitive wavelet coherenceanalysis, an increase in the coherence of 6 Hz oscillationsin the theta range is seen after the hypertonic stimulus,also a decrease in the coherence of delta oscillations ( < VIII. ACKNOWLEDGEMENTS
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