Research Article Analysis and Enhancements of a Prolific … · 2019. 7. 30. · of epilepsy that simulate the electroencephalogram (EEG) at the level of a neuronal population [ ]

Research ArticleAnalysis and Enhancements of a Prolific MacroscopicModel of Epilepsy

Christopher Fietkiewicz and Kenneth A Loparo

Electrical Engineering and Computer Science Department Case Western Reserve University Cleveland OH 44106 USA

Correspondence should be addressed to Christopher Fietkiewicz cxf47caseedu

Received 30 December 2015 Accepted 16 March 2016

Academic Editor Istvan Ulbert

Copyright copy 2016 C Fietkiewicz and K A Loparo This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Macroscopic models of epilepsy can deliver surprisingly realistic EEG simulations In the present study a prolific series of modelsis evaluated with regard to theoretical and computational concerns and enhancements are developed Specifically we analyze threeaspects of the models (1) Using dynamical systems analysis we demonstrate and explain the presence of direct current potentialsin the simulated EEG that were previously undocumented (2)We explain how the systemwas not ideally formulated for numericalintegration of stochastic differential equations A reformulated system is developed to support proper methodology (3)We explainan unreported contradiction in the published model specification regarding the use of a mathematical reduction method We thenuse the method to reduce the number of equations and further improve the computational efficiency The intent of our critique isto enhance the evolution of macroscopic modeling of epilepsy and assist others who wish to explore this exciting class of modelsfurther

1 Introduction

Significant attention has been given to computational modelsof epilepsy that simulate the electroencephalogram (EEG) atthe level of a neuronal population [1ndash4] Such models arereferred to by various names such as macroscopic neuralmass and mean field These models are capable of synthe-sizing realistic EEG time series with far less computationaleffort than that of microscopic models that operate at thescale of single neurons In addition to being efficient low-dimensional macroscopic models are also amenable to math-ematical analysis methods that can be used to understand keyproperties of the system being simulated

Most macroscopic models used in computational neu-roscience today are derived to some extent from one ofthree seminal formulations Freeman [5] Wilson and Cowan[6] and Lopes da Silva et al [7] In epilepsy modeling theapproach of Lopes da Silva et al is particularly prominentand has led to important hypotheses about epileptogenesisand the characteristics of the epileptiform EEG [4]

Wendling et al have been the most prolific in usingthe basic approach of Lopes da Silva with at least 17

different studies during the years 2000ndash2013 A key feature oftheir approach is the incorporation of synaptic interactionsbetween specific groups of neuronsThis permits the study ofa broad class of mechanisms for epileptogenesis that dependon the levels of network excitation and inhibition Most oftheir models are direct extensions of the previous work ofJansen et al [8 9] that modeled evoked response potentialsin human cortical columns

The earliest model of Wendling et al used the samestructure as Jansen et al and many of the same parame-ter values [10] Wendling et al qualitatively compared themodel to depth-EEG recordings from the human neocortexhippocampus and amygdala of patients with temporal lobeepilepsy (TLE) [10ndash17] Other models adhered to the samemethodology but increased the overall complexity in orderto achieve additional dynamical behaviors [18ndash26]

In the present work the modeling approach of Wendlinget al is critiquedwith regard to theoretical and computationalconcerns and enhancements are developed Specifically weanalyze three aspects of the models (1) Using dynamicalsystems analysis we demonstrate and explain the presenceof direct current potentials in the simulated EEG that were

Hindawi Publishing CorporationScientificaVolume 2016 Article ID 3628247 10 pageshttpdxdoiorg10115520163628247

2 Scientifica

previously undocumented (2) We explain how the systemwas not ideally formulated for numerical integration ofstochastic differential equations A reformulated system isdeveloped to support proper methodology (3)We explain anunreported contradiction in the published model specifica-tion regarding the use of a mathematical reduction methodWe then use the method to reduce the number of equationsand further improve the computational efficiency

2 Methods

21 Mathematical Model A basic diagram of the earliestmodel [10] is provided in Figure 1(a) and shows threeneuronal subgroups excitatory pyramidal cells excitatoryinterneurons and inhibitory interneuronsThe present studyused an extended version containing four subgroups [18] asshown in Figure 1(b) that contains an additional subgroup ofinhibitory interneurons

Figure 1(c) shows a detailed diagram of the specificcomputational components of the extended model A sub-group is a collection of similar but unconnected neuronsacting in parallel Inputs and outputs are represented asfiring rates or pulse densities For each subgroup twotypes of conversion blocks work together to transform theinput signal into an output signal The first is a pulse-to-voltage block that represents the process that occurs at aneuronal synapse An afferent pulse density is convertedto postsynaptic potentials (PSPs) that are either excitatory(EPSPs) or inhibitory (IPSPs) Mathematically this block isa linear transfer function implemented using a differentialequation The second conversion block is a voltage-to-pulseblock that translates the summed input voltages into asingle representative pulse density This block is a nonlinearalgebraic sigmoidal function

Subgroups are connected using multiplier constantslabeled 119862

1ndash1198627in Figure 1(c) These constants represent the

relative numbers of synaptic connections Parameters specificto each subgroup include average dendritic time constantsand average synaptic gains The synaptic gains correspond tothe relative magnitudes of PSPs and have typically been theonly parameters that were studied

The full equations are shown below using the originalvariable indexing [18] and Table 1 lists the model parameters

0(119905) = 119910

5 (1)

1(119905) = 119910

6 (2)

2(119905) = 119910

7 (3)

3(119905) = 119910

8 (4)

4(119905) = 119910

9 (5)

5(119905) = 119860119886119878 [119910

1(119905) minus 119910

2(119905) minus 119910

3(119905)] minus 2119886119910

5(119905)

minus 11988621199100(119905)

(6)

6(119905) = 119860119886 (119901 (119905) + 119862

2119878 [11986211199100(119905)]) minus 2119886119910

6(119905)

minus 11988621199101(119905)

(7)

7(119905) = 119861119887119862

4119878 [11986231199100(119905)] minus 2119887119910

7(119905) minus 119887

21199102(119905) (8)

8(119905) = 119866119892119862

7119878 [11986251199100(119905) minus 119862

61199104(119905)] minus 2119892119910

8(119905)

minus 11989221199103(119905)

(9)

9(119905) = 119861119887119878 [119862

31199100(119905)] minus 2119887119910

9(119905) minus 119887

21199104(119905) (10)

where 119901(119905) is a normally distributed random variable with amean of 90 and a variance of 30 119886 = 100 s 119887 = 50 s 119892 =

350 s 1198621= 135 119862

2= 08 times 119862

1 1198623= 1198624= 025 times 119862

1 1198625=

03 times 1198621 1198626= 01 times 119862

1 1198627= 08 times 119862

1 119878(V) = 2119890

0[1 +

119890119903(V0minusV)] V

0= 6mV 119890

0= 25 sminus1 119903 = 056mVminus1 119860 119861 and

119866 represent average synaptic gains the values of which arechosen to yield one of several possible types of model outputThe model output is defined as 119910out = 119910

1minus 1199102minus 1199103

Dynamical systems analysis was performed based on theabove equations using an approach described previously [27]for analysis of the initial model of Wendling et al [10] Tothe best of our knowledge the present study is the first toperform the analysis on the extendedmodel [18] Equilibriumpoints were solved numerically usingMathematica (WolframResearch Champaign Illinois) and the MATLAB Optimiza-tion Toolbox (The MathWorks Natick MA) Stability ofselected points was determined using the system Jacobianmatrix and computing the corresponding eigenvalues allusing the MATLAB Optimization Toolbox

22 Numerical Integration All simulations were performedusing MATLAB (The MathWorks Natick MA) Numeri-cal integration was done using a fixed-step forward Eulermethod Source code for simulations will be made avail-able publicly on ModelDB (httpssenselabmedyaleedumodeldb)

As noted in Results certain simulations used a stochasticforward Euler numerical integration method [28] The onlyequation that contains a random variable is (7) For thepurpose of stochastic numerical integration we redefined119901(119905) and (7) as follows

119901 (119905) = 119903 (119905)radicℎradic0001 (30) + 90

1199106(119905119899+1

) = 1199106(119905119899) + (119860119886) 119903 (119905

119899)radicℎradic0001 (30)

+ ℎ (119860119886 (90 + 1198622119878 [11986211199100(119905119899)]) minus 2119886119910

6(119905119899)

minus 11988621199101(119905119899))

(11)

where ℎ is the Euler integration step size and 119903(119905) is a normallydistributed random variable with a mean of 0 and a varianceof 1 The variable 119903(119905) is introduced so that ℎ can be seenas an explicit model parameter for a given simulation Wehave chosen a ldquoreferencerdquo step size of 0001 as used inprevious studies such that both the stochastic and classicalimplementations will be identical for ℎ = 0001 sec

Scientifica 3

P

I I

ExcitatoryInhibitory

r(t)

(a)

P

I II

r(t)


(b)

EPSP

IPSP

PyramidalExcitatory interneuronsInhibitory interneurons

IPSP

IPSP

EPSP

C1C2

C3C4

C5

C6

C7

y1

y2

y3

y4

y0

r(t)+

+

+

minusminus

+minus

p 997888rarr

p 997888rarr

p 997888rarr

p 997888rarr

p 997888rarr

997888rarr p

997888rarr p

997888rarr p

997888rarr p

(c)

Figure 1 Core models (a) Initial model [10] showing pyramidal (P) and interneuron (I) subgroups with either excitatory or inhibitoryprojections 119903(119905) is a random input (b) Extended model [18] (c) Detailed diagram of computational components for pyramidal cells (solidlines) excitatory interneurons (dotted lines) and inhibitory interneurons (dashed lines) Pulse-to-voltage blocks are labeled ldquo119901 rarr Vrdquo andvoltage-to-pulse blocks are labeled ldquoV rarr 119901rdquoThe IPSP block is distinguished from the EPSP blocks by awaveform showing negative deflection

Table 1 Model parameters fromWendling et al [18]

Param Interpretation Value119860 Average excitatory synaptic gain 5mV119861 Average slow inhibitory synaptic gain See Figure 2119866 Average fast inhibitory synaptic gain See Figure 2119886 Dendritic average time constant in the feedback excitatory loop 100 sminus1

119887 Dendritic average time constant in the slow feedback inhibitory loop 50 sminus1

119892 Somatic average time constant in the fast feedback inhibitory loop 350 sminus1

1198621 1198622 Mean number of synaptic contacts in the excitatory feedback loop 135 08 times 119862

1

1198623 1198624 Mean number of synaptic contacts in the slow feedback inhibitory loop 025 times 119862

1

1198625 1198626 Mean number of synaptic contacts in the fast feedback inhibitory loop 03 times 119862

1 01 times 119862

1

1198627

Mean number of synaptic contacts between slow and fast inhibitoryinterneurons 08 times 119862

1

3 Results

We analyze three aspects of the models (1) Using dynamicalsystems analysis we demonstrate and explain the presence ofdirect current potentials in the simulated EEG that were pre-viously undocumented (2) We explain how the system wasnot ideally formulated for numerical integration of stochasticdifferential equations A reformulated system is developed tosupport proper methodology (3) We explain an unreportedcontradiction in the published model specification regardingthe use of a mathematical reduction method We then usethe method to reduce the number of equations and furtherimprove the computational efficiency

31 DC Offset Models are not expected to be perfect replica-tions but knowledge of the underlying inaccuracies is critical

to proper usage [29] The model analyzed here exhibits anonzero mean potential that is different for each parameterconfiguration To the best of our knowledge this has neverbeen reported in any studies by Wendling et al This directcurrent (DC) offset is demonstrated in Figure 2 which showsa simulation similar to that of Wendling et al [18] Usingdifferent parameter combinations the simulation consistedof a progression of five phases of behavior (1) backgroundactivity (2) sporadic spiking (3) sustained spiking (4)gamma activity and (5) ictal activity For each phase thesimulation used a unique combination of values for theparameters 119861 and 119866 while 119860 remained fixed with a value of5 The inset of Figure 2 shows the values used for 119861 and 119866

The DC offset was present in a predecessor model [9]but did not appear in studies by Wendling et al that werebased on that modelThrough personal communication with

4 Scientifica

Simulated EEG Bifurcation diagram1 2 3 4 5

Phase12345

4538378

11

202020202

B G

B = 38

B = 8

B = 45

minus2

0

2

4

6

8

10

12

14

Out

put (

mV

)

5 10 15 20 25 30 35 40 450Time (sec)

minus2

0

2

4

6

8

10

12

14

you

t(m

V)

8 38 45 700Inhibitory gain (B)

Figure 2 Example of DC offset in the model (Left) Recreation of simulation of five different dynamical behaviors shown in Wendling et al[18] The inset shows values for parameters 119861 and 119866 that vary for each phase while 119860 is fixed at 5 (Right) A bifurcation diagram showingthe nullcline that predicts the DC offset based on the inhibitory gain 119861 for three different values (8 38 and 45) Filled circles are stableequilibrium points that correspond to similar mean values in the simulated EEG on the left (dashed lines) Open circles indicate bifurcationsbetween stability and instability on the nullclineThedark line between the open circles represents a continuous region of unstable equilibriumpoints

the author we learned that this DC potential was removedfrom simulations by either applying a high-pass filter orsubtracting the mean value

We used dynamical systems analysis to study theobserved DC offset values To the best of our knowledgethe present study is the first to perform such analysis on themodel For an accessible treatment of dynamical systems the-ory see Strogatz [30] For a specific discussion of the analysisof neural models see Milton et al [31] We computed theequilibrium points for Phases 1ndash4 where only the inhibitorygain 119861 was changed Phase 5 was not studied because itinvolved an additional parameter change for the inhibitorygain119866 However Phases 3 and 5 could be studied in a similarmanner

Using the approach of Grimbert and Faugeras [27] wederived the nullcline for the system = 119891(119884) = 0 Theequilibrium points were calculated by assigning = 0 andrearranging the resulting algebraic equations as follows

1199100=119860

119886119878 [1199101minus 1199102minus 1199103]

1199101=119860

119886(119901 + 119862

2119878 [11986211199100]) =

119860

119886(119901

+ 1198622119878 [1198621

119860

119886119878 [1199101minus 1199102minus 1199103]])

1199102=119861

1198871198624119878 [11986231199100] =

119861

1198871198624119878 [1198623

119860

119886119878 [1199101minus 1199102minus 1199103]]

1199103=119866

1198921198627119878 [11986251199100minus 11986261199104] =

119866

119892

sdot 1198627119878 [1198625

119860

119886119878 [1199101minus 1199102minus 1199103]

minus 1198626

119861

119887119878 [1198623

119860

119886119878 [1199101minus 1199102minus 1199103]]]

1199104=119861

119887119878 [11986231199100] =

119861

119887119878 [1198623

119860

119886119878 [1199101minus 1199102minus 1199103]]

(12)

Note that low-amplitude output fluctuations in the simula-tion are due to the random drive 119901(119905) For the dynamicalsystems analysis the mean of 119901(119905) was used (119901 = 90) Themodel output is defined as 119910out = 119910

1minus 1199102minus 1199103 Substituting

this into the equations above yields the following combinedequation

119910out

=119860

119886(119901 + 119862

2119878 [1198621

119860

119886119878 [119910out]])

minus119861

1198871198624119878 [1198623

119860

119886119878 [119910out]]

minus119866

1198921198627119878 [1198625

119860

119886119878 [119910out] minus 1198626

119861

119887119878 [1198623

119860

119886119878 [119910out]]]

(13)

Figure 2 (right) shows the nullcline for the system output119910out as determined by the above equation The nullclineincludes stable equilibrium points (filled circles) that cor-respond to similar mean values in the simulated EEG (seedashed lines) A continuous region of unstable equilibriumpoints (dark line) is also shown in Figure 2 between twobifurcations (open circles)

Table 2 lists the equilibrium points and their stabilitiesfor specific phases Stability is based on the eigenvalues ofthe system Jacobian matrix For stable points the real partsof all eigenvalues are negative For unstable points at leastone eigenvalue has a positive real part Tables 3 4 and 5 list

Scientifica 5

Table 2 Equilibrium points and their stabilities for specific phases119910out = 119910

1minus 1199102minus 1199103 For each equilibrium point 119910

5= 1199106= 1199107= 1199108=

1199109= 0 Stability is based on eigenvalues in Tables 3 4 and 5

Phase 119861 Stability 119910out 1199100

1199101

1199102

1199103

1199104

1 45Stable ndash0124 0008 6097 5882 0339 0174

Unstable 2526 0031 11777 8962 0290 0266Unstable 5087 0094 30864 25749 0028 0763

2 38Stable 1018 0014 7037 5600 0419 0166


3 37 Unstable 5466 0106 31254 25750 0037 07634 8 Stable 10004 0226 31500 19258 2238 0571

Table 3 Eigenvalues for Phase 1 119861 = 45

119910out minus0124 253 509Stability Stable Unstable Unstable1205820

minus1781 minus2414 minus1819

1205821


1205822


1205823


1205824

minus240 + 245119894 minus303 minus1344 + 989119894

1205825

minus3524 + 245119894 487 159 minus 788119894

1205826

minus240 minus 245119894 minus995 minus 1560119894 minus1344 minus 989119894

1205827


1205828

minus1017 minus 831119894 minus995 + 1560119894 159 + 788119894

1205829

minus1017 + 831119894 minus3587 + 427119894 minus3513 + 190119894


119910out 102 178 542Stability Stable Unstable Unstable1205820


1205821


1205822


1205823


1205824


1205825

minus142 minus 140119894 171 minus3515 minus 214119894

1205826


1205827

minus3552 minus 359119894 minus992 minus 1295119894 203 + 892119894

1205828

minus3552 + 359119894 minus992 + 1295119894 203 minus 892119894

1205829


the eigenvalues that correspond to the eight equilibriumpoints listed in Table 2The eigenvalues were computed fromthe system Jacobianmatrix using theMATLAB Symbolic andOptimization Toolboxes (The MathWorks Natick MA) Forall tables 119910out corresponds to the ordinate (vertical axis) inFigure 2 (right) and can be used to identify each uniqueequilibrium point For stable points the real parts of alleigenvalues are negative For unstable points at least oneeigenvalue has a positive real part

In Figure 2 and Table 2 it can be seen that Phases 1 and 2each have one stable and two unstable equilibrium points As

Table 5 Eigenvalues for Phase 3 (119861 = 37) and Phase 4 (119861 = 8)

Phase 3119861 = 37

Phase 4119861 = 8

119910out 547 1000Stability Stable Unstable1205820 minus1378 minus1720

1205821 minus847 minus593



1205824 minus1579 + 919119894 minus3522 + 236119894

1205825 minus3516 minus 219119894 minus326 + 98119894

1205826 minus1579 minus 919119894 minus3522 minus 236119894

1205827 207 + 902119894 minus995 minus 771119894

1205828 207 minus 902119894 minus326 minus 98119894

1205829 minus3516 + 219119894 minus995 + 771119894

119861 decreases a bifurcation occurs at 119861 = 373 where a stablepoint and one unstable point both disappear This saddle-node bifurcation corresponds to the emergence of a stablelimit cycle that is present in Phase 3 where one unstableequilibrium point remains As 119861 decreases further anotherbifurcation occurs at which the limit cycle disappears andone stable equilibrium point remains We determined thatthis bifurcation occurs in the range 921 lt 119861 lt 922 basedon a systematic stability analysis of equilibrium points in thatregion

Of particular interest in the above analysis is the DCoffset that is different for each phase High-pass filtering iscommonly used in EEG acquisition to eliminate DC andimprove dynamic range DC offset in the physiological EEGis still poorly understood but studies have shown that itcoincides with ictal activity in some circumstances Contraryto the simulation in Figure 2 most studies show that theshift is invariably negative with respect to the baseline [32ndash35] though the opposite has also been observed [36] Of thesestudies the maximum shift reported was 23mV in contrastto a shift of 10mV in the simulations

Clearly the model was not designed to accurately sim-ulate the DC shift because electrochemical effects are notdirectly accounted for Electrochemical changes are the mostlikely mechanism responsible for the DC offset that isseen in the EEG To the best of our knowledge only oneepilepsy model has specifically addressed EEG offset [34 35]However that model is not based on physiology and doesnot specifically account for electrochemical dynamics In factthe authors state that the model output was circumstantiallychosen as the sum of two state variables because of its visualresemblance to the EEG [35] Furthermore the study does notsuggest any physiological significance of the DC offset

Note that Labyt et al [22] described studying whatthey called ldquostabilityrdquo properties for a significantly morecomplex model with twelve neuronal subgroups Howeverthat analysis was a Monte Carlo procedure in which the termldquostablerdquo was used to distinguish nonictal behavior from ictal

6 Scientifica

05 10Time (sec)

minus05

0

05

Out

put (

mV

)

(a)

minus05

0

05

Out

put (

mV

)05 10

Time (sec)

(b)

minus05

0

05

Out

put (

mV

)

05 10Time (sec)

(c)

Figure 3 Examples of using the 4th-order Runge-Kutta method as in Wendling et al [18] (a) For integration step size of 1ms the variance(1205902) of the output is 00751 (b) For integration step size of 01ms 1205902 is 00065 (c) For integration step size of 001ms 1205902 is 00006 Varianceswere computed using 10 seconds of output Model parameters 119860 = 5 119861 = 40 and 119866 = 20

behavior It did not address the DC offset and it was not ananalysis of dynamical stability

32 Numerical Integration The literature indicates that themodel has historically been simulated as a set of ordinarydifferential equations This was verified by source code thatis publicly available on ModelDB (httpssenselabmedyaleedumodeldb) Aproblemarises because the systemactuallyconsists of stochastic differential equations (SDEs) due tothe inclusion of the random variable 119901(119905) in (7) Classicalnumerical integration methods are not appropriate for SDEsbecause they scale random variables by the integration stepsize For a practical introduction to SDEs see [28 37 38]

An undesirable consequence is that as the integrationstep size becomes smaller (eg 1ms 01ms and 001ms) theaccuracy of the simulated output actually decreases Figure 3shows examples of simulating the model with a 4th-orderRunge-Kutta method as described in Wendling et al [18]using different integration step sizes Notice that the relativepeak-to-peak amplitude and variance of the signal continueto decrease as the integration step size decreases instead ofconverging to stable values The variance actually changesby the same order of magnitude as the step size suggestinga direct relationship between the two The graphs show 1-second simulations in order to visually compare the similarityin waveform shapes However the variances were calculatedusing 10-second simulations in order to obtain values thatwere consistent across multiple simulations for the same stepsize

We addressed the issue by reformulating (7) such that thestochastic term is not scaled by the integration step size (seeSection 2) Figure 4 shows examples of using this approachUsing this revised formulation there is still a slight decreasein variance but it is greatly improved in comparison toFigure 3 Note that each simulation in Figure 4 is a uniquesolution because each change in step size involves a different

number of random input samples For a step size of 1ms thesimulations in Figures 3 and 4 are identical This is due to acareful reformulation of the equations prior to implementingthe SDE numerical method In the following section we willprovide the reformulation in the context of the full set ofequations Technically all of the simulations published byWendling et al could be duplicated using this approach

33 Equation Reduction Lastly we present a discrepancy inthe published models that to the best of our knowledgehas never been addressed In Wendling et al [18] thestructure of themodel can be difficult to interpret because themathematical definition strayed from the actual physiologybeing modeled As described in Methods each PSP blocktranslates into one differential equation A careful readingof the 2002 article reveals seven PSP blocks but only fivedifferential equations The equations actually agreed with anearlier version of the block diagram in Wendling et al [10]but that diagram was not consistent with the physiologicalinterpretation of the PSP block Ironically that diagram wasbased on earlier studies that also had the same inconsistency[8 9] The change that occurred in these earlier studies wasthat the two conversion blocks within each subgroup wereartificially separatedThis simplified themodel such thatmul-tiple subgroups could share the same block Specifically theexcitatory interneuron group and the inhibitory interneurongroup were revised so as to share the same PSP conversionblock Figure 5 compares a physiologically accurate blockdiagram with the modified version

The reduction is reasonable considering that both sub-groups share the same input and that the PSP block ismodeled as a linear transfer function The end result is areduction in the required number of equations For unknownreasons a similar reductionwas not applied to the dual outputpaths from inhibitory interneuronswhose outputwas definedas 1199104 Such a reduction would increase the efficiency of

Scientifica 7

minus05

0

05

Out

put (

mV

)

05 10Time (sec)

(a)

minus05

0

05

Out

put (

mV

)05 10

Time (sec)

(b)

minus05

0

05

Out

put (

mV

)

05 10Time (sec)

(c)

Figure 4 Examples of using forward Euler for SDEs (a) For integration step size of 1ms the variance (1205902) of the output is 00779 (b) Forintegration step size of 01ms 1205902 is 00655 (c) For integration step size of 001ms 1205902 is 00621 Variances were computed using 10 seconds ofoutput Model parameters 119860 = 5 119861 = 40 and 119866 = 20

Pyramidal cells

EPSP

EPSP

C1C2

C3C4

p 997888rarr

p 997888rarr

997888rarr p

997888rarr p

(a)

EPSPPyramidal cells

C1C2

C3C4

p 997888rarr

997888rarr p

997888rarr p

(b)

Figure 5Mathematically equivalentmodels (a) Jansen et al [8] are physiologically accuratewith one PSP block for each input into a neuronalsubgroup The same approach was used in Wendling et al but contradicts the equations (b) The structure was changed in Jansen and Rit[9] such that two neuronal subgroups share a common PSP block Cell groups pyramidal cells (solid lines) excitatory interneurons (dottedlines) and inhibitory interneurons (dashed lines) The EPSP block is shown with dash-dotted lines to indicate that it corresponds to both theexcitatory and inhibitory EPSP blocks in (a)

the formulation further Figure 6 shows this reduction inwhich the former 119910

4has been renamed as 119910

2 and the IPSP

block for the former 1199102has been removed

A reduced set of equations is provided below that containsthree major revisions First the input to the pyramidal cellsnow uses the multiplier 119862

4(see the equation for the new

4)

Second the variables formerly named 1199104and 119910

9have been

renamed as 1199102and 119910

6 respectively Third we have separated

the stochastic and deterministic terms to enable the properuse of numerical integrationmethods as described earlier (seethe equation for the new

5) The complete system is defined

as follows

0(119905) = 119910

4 (14)

1(119905) = 119910

5 (15)

2(119905) = 119910

6 (16)

3(119905) = 119910

7 (17)

4(119905) = 119860119886119878 [119910

1(119905) minus 119862

41199102(119905) minus 119910

3(119905)] minus 2119886119910

4(119905)

minus 11988621199100(119905)

(18)

5(119905) = radic0001119860119886120590

2119903 (119905)

+ (119860119886 (119898 + 1198622119878 [11986211199100(119905)]) minus 2119886119910

5(119905) minus 119886

21199101(119905))

(19)

6(119905) = 119861119887119878 [119862

31199100(119905)] minus 2119887119910

6(119905) minus 119887

21199102(119905) (20)

7(119905) = 119866119892119862

7119878 [11986251199100(119905) minus 119862

61199104(119905)] minus 2119892119910

7(119905)

minus 11989221199103(119905)

(21)

where 119903(119905) is a normally distributed random variable with amean of 0 and a variance of 1 1205902 is the desired input variance

8 Scientifica

EPSP

IPSP

IPSP

EPSP

C1C2

C3C4

C5

C6

C7

r(t) y1

y2

y3

y0

+

+

++

minus

minusminus


p 997888rarr

p 997888rarr

p 997888rarr

p 997888rarr

997888rarr p

997888rarr p

997888rarr p

997888rarr p

Figure 6 Revised model with additional mathematical reduction The former 1199104has been renamed as 119910

2 and the IPSP block for the former

1199102has been removed

and119898 is the desired input mean We repeated the simulationshown in Figure 2 and confirmed that the results are identical

4 Discussion

We critiqued a prolific computational modeling approachthat has been used for the study of epilepsy We evaluatedthree aspects of the models with regard to theoretical andcomputational concerns and we developed enhancements tothe model formulation None of the issues that were raisedinvalidate the published results However we feel they areimportant considerations for other researchers to utilize themodels effectively

First we demonstrated that a previously unreported DCoffset is present in the model and that the offset varies fordifferent parameter configurations As explained previouslythe presence of a DC offset is a well-known characteristic ofthe physiological EEG that is typically ignored However themodel was not designed to incorporate any supposed mech-anism for this phenomenon Though the model produces anoutput that is interpreted as a voltage the reactive-diffusiveprocess of extracellular ion flow is in no way described by thesystemWe used dynamical systems analysis to show how theDC offset in the model can be predicted from the equationsThough another model has specifically addressed DC offset[35] no physiological significance was suggested Futurework can explore whether the model correctly describes thephenomenon that is observed in physiological systems

Second we described how numerical integration meth-ods may significantly affect the results Using the pub-lished method the voltage amplitude of the simulated EEGwas greatly affected by the integration step size Methodsappropriate for SDEs require a separation of stochastic anddeterministic terms From a practical perspective this affectswhether results are reproducible by other researchers Weprovided a reformulation of the equations in order to separate

the stochastic and deterministic terms and we describedhow this formulation would be implemented using a forwardEuler integration method

Note that there are additional numerical methods avail-able for SDEs For example a stochastic Runge-Kuttamethodexists [28] but it is only applicable when the random variableis multiplicative with respect to a state variable In thepresent system the term with the random variable does notcontain a state variable Two significantly different integrationapproaches can be found in [39 40] The latter study isactually based on the EEG model in [9] However theseapproaches cannot be compared directly to classical methodsin the same manner as we have done here Future workcan evaluate the efficiency of these alternative integrationmethods for the present model

Third we discussed a mathematical reduction that led toa contradiction between system diagrams and the equationsin the literature We further proposed a modification toimprove the efficiency of the equations by applying the samemathematical reduction to a different part of the modelThough the reduction is mathematically equivalent to thelonger form it is an important conceptual modificationbecause it contradicts actual physiological structure

The intent of our critique is to enhance the evolutionof macroscopic modeling of epilepsy and assist others whowish to explore this exciting class of models further Just asmodeling is only one research tool amongmanymacroscopicmodeling is merely one of many levels of modeling thatare needed to study a system like the brain that exhibitscomplexities at many temporal and spatial scales Micro-scopic models of large networks may require significantcomputing power but macroscopic models can usually besimulated using common office computing equipment Aswe have demonstrated here low-dimensional models alsoallow for rigorous mathematical analysis in order to betterunderstand the mechanisms behind dynamical behavior

Scientifica 9

These advantages can benefit epilepsy research as well asneuroscience in general

Additional Points

Source code for simulations will be made available publiclyon ModelDB (httpssenselabmedyaleedumodeldb)

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

The authors thank the reviewers for helpful suggestions

References

[1] P Suffczynski F Wendung J-J Bellanger and F H LopesDa Silva ldquoSome insights into computational models of(patho)physiological brain activityrdquoProceedings of the IEEE vol94 no 4 pp 784ndash804 2006

[2] FWendling F Bartolomei F Mina C Huneau and P BenquetldquoInterictal spikes fast ripples and seizures in partial epilepsiesmdashcombining multi-level computational models with experimen-tal datardquo European Journal of Neuroscience vol 36 no 2 pp2164ndash2177 2012

[3] J G R Jefferys L Menendez de la Prida F Wendling et alldquoMechanisms of physiological and epileptic HFO generationrdquoProgress in Neurobiology vol 98 no 3 pp 250ndash264 2012

[4] F Wendling P Benquet F Bartolomei and V Jirsa ldquoComputa-tional models of epileptiform activityrdquo Journal of NeuroscienceMethods vol 260 pp 233ndash251 2016

[5] W J Freeman ldquoA linear distributed feedbackmodel for prepyri-form cortexrdquoExperimentalNeurology vol 10 no 6 pp 525ndash5471964

[6] H R Wilson and J D Cowan ldquoExcitatory and inhibitory inter-actions in localized populations of model neuronsrdquo BiophysicalJournal vol 12 no 1 pp 1ndash24 1972

[7] F H Lopes da Silva A van Rotterdam P Barts E vanHeusden and W Burr ldquoModels of neuronal populations thebasic mechanisms of rhythmicityrdquo Progress in Brain Researchvol 45 pp 281ndash308 1976

[8] B H Jansen G Zouridakis and M E Brandt ldquoAneurophysiologically-based mathematical model of flashvisual evoked potentialsrdquo Biological Cybernetics vol 68 no 3pp 275ndash283 1993

[9] B H Jansen and V G Rit ldquoElectroencephalogram and visualevoked potential generation in a mathematical model of cou-pled cortical columnsrdquo Biological Cybernetics vol 73 no 4 pp357ndash366 1995

[10] F Wendling J J Bellanger F Bartolomei and P ChauvelldquoRelevance of nonlinear lumped-parameter models in theanalysis of depth-EEG epileptic signalsrdquo Biological Cyberneticsvol 83 no 4 pp 367ndash378 2000

[11] F Wendling and F Bartolomei ldquoModeling EEG signals andinterpreting measures of relationship during temporal-lobeseizures an approach to the study of epileptogenic networksrdquoEpileptic Disorders vol 3 no 1 pp 67ndash78 2001

[12] F Wendling J R L Webb F Bartolomei J J Bellanger andP Chauvel ldquoInterpretation of interdependencies in epilepticsignals using a macroscopic physiological model of the EEGrdquoClinical Neurophysiology vol 112 no 7 pp 1201ndash1218 2001

[13] F Wendling F Bartolomei J J Bellanger and P ChauvelldquoIdentification of epileptogenic networks from modeling andnonlinear analysis of SEEG signalsrdquo Neurophysiologie Cliniquevol 31 no 3 pp 139ndash151 2001

[14] F Bartolomei F Wendling J-J Bellanger J Regis and PChauvel ldquoNeural networks involving the medial temporalstructures in temporal lobe epilepsyrdquo Clinical Neurophysiologyvol 112 no 9 pp 1746ndash1760 2001

[15] D Cosandier-Rimele J-M Badier and FWendling ldquoA realisticspatiotemporal source model for EEG activity application tothe reconstruction of epileptic depth-EEG signalsrdquo in Proceed-ings of the 28th Annual International Conference of the IEEEEngineering in Medicine and Biology Society (EMBS rsquo06) pp4253ndash4256 IEEE New York NY USA September 2006

[16] D Cosandier-Rimele J-M Badier P Chauvel and FWendlingldquoA physiologically plausible spatio-temporal model for EEGsignals recorded with intracerebral electrodes in human partialepilepsyrdquo IEEE Transactions on Biomedical Engineering vol 54no 3 pp 380ndash388 2007

[17] C Huneau P Benquet G Dieuset A Biraben B Martin andF Wendling ldquoShape features of epileptic spikes are a marker ofepileptogenesis inmicerdquo Epilepsia vol 54 no 12 pp 2219ndash22272013

[18] F Wendling F Bartolomei J J Bellanger and P ChauvelldquoEpileptic fast activity can be explained by a model of impairedGABAergic dendritic inhibitionrdquo European Journal of Neuro-science vol 15 no 9 pp 1499ndash1508 2002

[19] F Wendling A Hernandez J-J Bellanger P Chauvel and FBartolomei ldquoInterictal to ictal transition in human temporallobe epilepsy insights from a computational model of intrac-erebral EEGrdquo Journal of Clinical Neurophysiology vol 22 no 5pp 343ndash356 2005

[20] E Labyt L UvaMDeCurtis and FWendling ldquoRealisticmod-eling of entorhinal cortex field potentials and interpretation ofepileptic activity in the guinea pig isolated brain preparationrdquoJournal of Neurophysiology vol 96 no 1 pp 363ndash377 2006

[21] L El-hassarMMilh FWendling N FerrandM Esclapez andC Bernard ldquoCell domain-dependent changes in the glutamater-gic and GABAergic drives during epileptogenesis in the rat CA1regionrdquo Journal of Physiology vol 578 no 1 pp 193ndash211 2007

[22] E Labyt P Frogerais L Uva J-J Bellanger and F WendlingldquoModeling of entorhinal cortex and simulation of epilepticactivity insights into the role of inhibition-related parametersrdquoIEEE Transactions on Information Technology in Biomedicinevol 11 no 4 pp 450ndash461 2007

[23] B Molaee-Ardekani P Benquet F Bartolomei and FWendling ldquoComputational modeling of high-frequencyoscillations at the onset of neocortical partial seizures fromlsquoaltered structurersquo to lsquodysfunctionrsquordquo NeuroImage vol 52 no 3pp 1109ndash1122 2010

[24] I Merlet G Birot R Salvador et al ldquoFrom oscillatory transcra-nial current stimulation to scalp EEG changes a biophysical andphysiological modeling studyrdquo PLoS ONE vol 8 no 2 ArticleID e57330 2013

[25] F Mina P Benquet A Pasnicu A Biraben and F WendlingldquoModulation of epileptic activity by deep brain stimulation amodel-based study of frequency-dependent effectsrdquo Frontiers inComputational Neuroscience vol 7 article 94 2013

10 Scientifica

[26] B Molaee-Ardekani J Marquez-Ruiz I Merlet et al ldquoEffectsof transcranial Direct Current Stimulation (tDCS) on corticalactivity a computational modeling studyrdquo Brain Stimulationvol 6 no 1 pp 25ndash39 2013

[27] F Grimbert and O Faugeras ldquoBifurcation analysis of Jansenrsquosneural mass modelrdquo Neural Computation vol 18 no 12 pp3052ndash3068 2006

[28] S Cyganowski P Kloeden and J Ombach From ElementaryProbability to Stochastic Differential Equations with MapleSpringer Berlin Germany 2002

[29] J M Bower and C Koch ldquoExperimentalists and modelers canwe all just get alongrdquo Trends in Neurosciences vol 15 no 11 pp458ndash461 1992

[30] S Strogatz Nonlinear Dynamics and Chaos With Applicationsto Physics Biology Chemistry and Engineering Westview PressBoulder Colo USA 2000

[31] J G Milton J Foss J D Hunter and J L Cabrera ldquoControllingneurological disease at the edge of instabilityrdquo in QuantitativeNeuroscience Models Algorithms Diagnostics and TherapeuticApplications P M Pardalos Ed pp 117ndash143 Kluwer AcademicPublishers Boston Mass USA 2004

[32] E J Speckmann and C E Elger ldquoIntroduction to the neu-rophysiological basis of the EEG and DC potentialsrdquo in Elec-troencephalography Basic Principles Clinical Applications andRelated Fields E Niedermeyer and FH Lopes da Silva Eds pp15ndash27 Lippincott Williams amp Wilkins Philadelphia Pa USA2005

[33] S Vanhatalo J Voipio and K Kaila ldquoInfraslow EEG activityrdquo inElectroencephalography Basic Principles Clinical Applicationsand Related Fields E Niedermeyer and F H Lopes da SilvaEds pp 489ndash493 LippincottWilliamsampWilkins PhiladelphiaPa USA 5th edition 2005

[34] V K Jirsa W C Stacey P P Quilichini A I Ivanov and CBernard ldquoOn the nature of seizure dynamicsrdquo Brain vol 137no 8 pp 2210ndash2230 2014

[35] T Proix F Bartolomei P Chauvel C Bernard and V K JirsaldquoPermittivity coupling across brain regions determines seizurerecruitment in partial epilepsyrdquo Journal of Neuroscience vol 34no 45 pp 15009ndash15021 2014

[36] E C Mader Jr B J Fisch M E Carey and N R Villemarette-Pittman ldquoIctal onset slow potential shifts recorded with hip-pocampal depth electrodesrdquo Neurology amp Clinical Neurophys-iology vol 2005 no 4 2005

[37] D J Higham ldquoAn algorithmic introduction to numericalsimulation of stochastic differential equationsrdquo SIAM Reviewvol 43 no 3 pp 525ndash546 2001

[38] K Burrage I Lenane and G Lythe ldquoNumerical methods forsecond-order stochastic differential equationsrdquo SIAM Journalon Scientific Computing vol 29 no 1 pp 245ndash264 2007

[39] L Grune and P E Kloeden ldquoPathwise approximation ofrandom ordinary differential equationsrdquo BIT Numerical Math-ematics vol 41 no 4 pp 711ndash721 2001

[40] R C Sotero N J Trujillo-Barreto Y Iturria-Medina F Car-bonell and J C Jimenez ldquoRealistically coupled neural massmodels can generate EEG rhythmsrdquo Neural Computation vol19 no 2 pp 478ndash512 2007

Submit your manuscripts athttpwwwhindawicom

Neurology Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Alzheimerrsquos DiseaseHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


BioMed Research International


Research and TreatmentSchizophrenia

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Neural Plasticity


Parkinsonrsquos Disease


Research and TreatmentAutism

Sleep DisordersHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Neuroscience Journal

Epilepsy Research and TreatmentHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Psychiatry Journal


Computational and Mathematical Methods in Medicine

Depression Research and TreatmentHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Brain ScienceInternational Journal of

StrokeResearch and TreatmentHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Neurodegenerative Diseases


Journal of

Cardiovascular Psychiatry and NeurologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

2 Scientifica

previously undocumented (2) We explain how the systemwas not ideally formulated for numerical integration ofstochastic differential equations A reformulated system isdeveloped to support proper methodology (3)We explain anunreported contradiction in the published model specifica-tion regarding the use of a mathematical reduction methodWe then use the method to reduce the number of equationsand further improve the computational efficiency

2 Methods

21 Mathematical Model A basic diagram of the earliestmodel [10] is provided in Figure 1(a) and shows threeneuronal subgroups excitatory pyramidal cells excitatoryinterneurons and inhibitory interneuronsThe present studyused an extended version containing four subgroups [18] asshown in Figure 1(b) that contains an additional subgroup ofinhibitory interneurons

Figure 1(c) shows a detailed diagram of the specificcomputational components of the extended model A sub-group is a collection of similar but unconnected neuronsacting in parallel Inputs and outputs are represented asfiring rates or pulse densities For each subgroup twotypes of conversion blocks work together to transform theinput signal into an output signal The first is a pulse-to-voltage block that represents the process that occurs at aneuronal synapse An afferent pulse density is convertedto postsynaptic potentials (PSPs) that are either excitatory(EPSPs) or inhibitory (IPSPs) Mathematically this block isa linear transfer function implemented using a differentialequation The second conversion block is a voltage-to-pulseblock that translates the summed input voltages into asingle representative pulse density This block is a nonlinearalgebraic sigmoidal function

Subgroups are connected using multiplier constantslabeled 119862

1ndash1198627in Figure 1(c) These constants represent the

relative numbers of synaptic connections Parameters specificto each subgroup include average dendritic time constantsand average synaptic gains The synaptic gains correspond tothe relative magnitudes of PSPs and have typically been theonly parameters that were studied

The full equations are shown below using the originalvariable indexing [18] and Table 1 lists the model parameters

0(119905) = 119910

5 (1)

1(119905) = 119910

6 (2)

2(119905) = 119910

7 (3)

3(119905) = 119910

8 (4)

4(119905) = 119910

9 (5)

5(119905) = 119860119886119878 [119910

1(119905) minus 119910

2(119905) minus 119910

3(119905)] minus 2119886119910

5(119905)

minus 11988621199100(119905)

(6)

6(119905) = 119860119886 (119901 (119905) + 119862

2119878 [11986211199100(119905)]) minus 2119886119910

6(119905)

minus 11988621199101(119905)

(7)

7(119905) = 119861119887119862

4119878 [11986231199100(119905)] minus 2119887119910

7(119905) minus 119887

21199102(119905) (8)

8(119905) = 119866119892119862

7119878 [11986251199100(119905) minus 119862

61199104(119905)] minus 2119892119910

8(119905)

minus 11989221199103(119905)

(9)

9(119905) = 119861119887119878 [119862

31199100(119905)] minus 2119887119910

9(119905) minus 119887

21199104(119905) (10)

where 119901(119905) is a normally distributed random variable with amean of 90 and a variance of 30 119886 = 100 s 119887 = 50 s 119892 =

350 s 1198621= 135 119862

2= 08 times 119862

1 1198623= 1198624= 025 times 119862

1 1198625=

03 times 1198621 1198626= 01 times 119862

1 1198627= 08 times 119862

1 119878(V) = 2119890

0[1 +

119890119903(V0minusV)] V

0= 6mV 119890

0= 25 sminus1 119903 = 056mVminus1 119860 119861 and

119866 represent average synaptic gains the values of which arechosen to yield one of several possible types of model outputThe model output is defined as 119910out = 119910

1minus 1199102minus 1199103

Dynamical systems analysis was performed based on theabove equations using an approach described previously [27]for analysis of the initial model of Wendling et al [10] Tothe best of our knowledge the present study is the first toperform the analysis on the extendedmodel [18] Equilibriumpoints were solved numerically usingMathematica (WolframResearch Champaign Illinois) and the MATLAB Optimiza-tion Toolbox (The MathWorks Natick MA) Stability ofselected points was determined using the system Jacobianmatrix and computing the corresponding eigenvalues allusing the MATLAB Optimization Toolbox

22 Numerical Integration All simulations were performedusing MATLAB (The MathWorks Natick MA) Numeri-cal integration was done using a fixed-step forward Eulermethod Source code for simulations will be made avail-able publicly on ModelDB (httpssenselabmedyaleedumodeldb)

As noted in Results certain simulations used a stochasticforward Euler numerical integration method [28] The onlyequation that contains a random variable is (7) For thepurpose of stochastic numerical integration we redefined119901(119905) and (7) as follows

119901 (119905) = 119903 (119905)radicℎradic0001 (30) + 90

1199106(119905119899+1

) = 1199106(119905119899) + (119860119886) 119903 (119905

119899)radicℎradic0001 (30)

+ ℎ (119860119886 (90 + 1198622119878 [11986211199100(119905119899)]) minus 2119886119910

6(119905119899)

minus 11988621199101(119905119899))

(11)

where ℎ is the Euler integration step size and 119903(119905) is a normallydistributed random variable with a mean of 0 and a varianceof 1 The variable 119903(119905) is introduced so that ℎ can be seenas an explicit model parameter for a given simulation Wehave chosen a ldquoreferencerdquo step size of 0001 as used inprevious studies such that both the stochastic and classicalimplementations will be identical for ℎ = 0001 sec

Scientifica 3

P

I I


r(t)

(a)

P

I II

r(t)


(b)

EPSP

IPSP


IPSP

IPSP

EPSP

C1C2

C3C4

C5

C6

C7

y1

y2

y3

y4

y0

r(t)+

+

+

minusminus

+minus

p 997888rarr

p 997888rarr

p 997888rarr

p 997888rarr

p 997888rarr

997888rarr p

997888rarr p

997888rarr p

997888rarr p

(c)







1


1


1 01 times 119862

1

1198627


1

3 Results





4 Scientifica


Phase12345

4538378

11

202020202

B G

B = 38

B = 8

B = 45

minus2

0

2

4

6

8

10

12

14

Out

put (

mV

)

5 10 15 20 25 30 35 40 450Time (sec)

minus2

0

2

4

6

8

10

12

14

you

t(m

V)






1199100=119860

119886119878 [1199101minus 1199102minus 1199103]

1199101=119860

119886(119901 + 119862

2119878 [11986211199100]) =

119860

119886(119901

+ 1198622119878 [1198621

119860

119886119878 [1199101minus 1199102minus 1199103]])

1199102=119861

1198871198624119878 [11986231199100] =

119861

1198871198624119878 [1198623

119860

119886119878 [1199101minus 1199102minus 1199103]]

1199103=119866

1198921198627119878 [11986251199100minus 11986261199104] =

119866

119892

sdot 1198627119878 [1198625

119860

119886119878 [1199101minus 1199102minus 1199103]

minus 1198626

119861

119887119878 [1198623

119860

119886119878 [1199101minus 1199102minus 1199103]]]

1199104=119861

119887119878 [11986231199100] =

119861

119887119878 [1198623

119860

119886119878 [1199101minus 1199102minus 1199103]]

(12)




119910out

=119860

119886(119901 + 119862

2119878 [1198621

119860

119886119878 [119910out]])

minus119861

1198871198624119878 [1198623

119860

119886119878 [119910out]]

minus119866

1198921198627119878 [1198625

119860

119886119878 [119910out] minus 1198626

119861

119887119878 [1198623

119860

119886119878 [119910out]]]

(13)



Scientifica 5



5= 1199106= 1199107= 1199108=



1199101

1199102

1199103

1199104

1 45Stable ndash0124 0008 6097 5882 0339 0174


2 38Stable 1018 0014 7037 5600 0419 0166


3 37 Unstable 5466 0106 31254 25750 0037 07634 8 Stable 10004 0226 31500 19258 2238 0571




1205821


1205822


1205823


1205824


1205825

minus3524 + 245119894 487 159 minus 788119894

1205826


1205827


1205828


1205829





1205821


1205822


1205823


1205824


1205825


1205826


1205827


1205828


1205829





Phase 3119861 = 37

Phase 4119861 = 8





1205824 minus1579 + 919119894 minus3522 + 236119894



1205827 207 + 902119894 minus995 minus 771119894


1205829 minus3516 + 219119894 minus995 + 771119894





6 Scientifica

05 10Time (sec)

minus05

0

05

Out

put (

mV

)

(a)

minus05

0

05

Out

put (

mV

)05 10

Time (sec)

(b)

minus05

0

05

Out

put (

mV

)

05 10Time (sec)

(c)









Scientifica 7

minus05

0

05

Out

put (

mV

)

05 10Time (sec)

(a)

minus05

0

05

Out

put (

mV

)05 10

Time (sec)

(b)

minus05

0

05

Out

put (

mV

)

05 10Time (sec)

(c)


Pyramidal cells

EPSP

EPSP

C1C2

C3C4

p 997888rarr

p 997888rarr

997888rarr p

997888rarr p

(a)

EPSPPyramidal cells

C1C2

C3C4

p 997888rarr

997888rarr p

997888rarr p

(b)




2 and the IPSP




4)


9have been





as follows

0(119905) = 119910

4 (14)

1(119905) = 119910

5 (15)

2(119905) = 119910

6 (16)

3(119905) = 119910

7 (17)

4(119905) = 119860119886119878 [119910

1(119905) minus 119862

41199102(119905) minus 119910

3(119905)] minus 2119886119910

4(119905)

minus 11988621199100(119905)

(18)

5(119905) = radic0001119860119886120590

2119903 (119905)

+ (119860119886 (119898 + 1198622119878 [11986211199100(119905)]) minus 2119886119910

5(119905) minus 119886

21199101(119905))

(19)

6(119905) = 119861119887119878 [119862

31199100(119905)] minus 2119887119910

6(119905) minus 119887

21199102(119905) (20)

7(119905) = 119866119892119862

7119878 [11986251199100(119905) minus 119862

61199104(119905)] minus 2119892119910

7(119905)

minus 11989221199103(119905)

(21)


8 Scientifica

EPSP

IPSP

IPSP

EPSP

C1C2

C3C4

C5

C6

C7

r(t) y1

y2

y3

y0

+

+

++

minus

minusminus


p 997888rarr

p 997888rarr

p 997888rarr

p 997888rarr

997888rarr p

997888rarr p

997888rarr p

997888rarr p





4 Discussion








Scientifica 9


Additional Points


Competing Interests


Acknowledgments


References


























10 Scientifica




























Neural Plasticity










Psychiatry Journal









Journal of


Scientifica 3

P

I I


r(t)

(a)

P

I II

r(t)


(b)

EPSP

IPSP


IPSP

IPSP

EPSP

C1C2

C3C4

C5

C6

C7

y1

y2

y3

y4

y0

r(t)+

+

+

minusminus

+minus

p 997888rarr

p 997888rarr

p 997888rarr

p 997888rarr

p 997888rarr

997888rarr p

997888rarr p

997888rarr p

997888rarr p

(c)







1


1


1 01 times 119862

1

1198627


1

3 Results





4 Scientifica


Phase12345

4538378

11

202020202

B G

B = 38

B = 8

B = 45

minus2

0

2

4

6

8

10

12

14

Out

put (

mV

)

5 10 15 20 25 30 35 40 450Time (sec)

minus2

0

2

4

6

8

10

12

14

you

t(m

V)






1199100=119860

119886119878 [1199101minus 1199102minus 1199103]

1199101=119860

119886(119901 + 119862

2119878 [11986211199100]) =

119860

119886(119901

+ 1198622119878 [1198621

119860

119886119878 [1199101minus 1199102minus 1199103]])

1199102=119861

1198871198624119878 [11986231199100] =

119861

1198871198624119878 [1198623

119860

119886119878 [1199101minus 1199102minus 1199103]]

1199103=119866

1198921198627119878 [11986251199100minus 11986261199104] =

119866

119892

sdot 1198627119878 [1198625

119860

119886119878 [1199101minus 1199102minus 1199103]

minus 1198626

119861

119887119878 [1198623

119860

119886119878 [1199101minus 1199102minus 1199103]]]

1199104=119861

119887119878 [11986231199100] =

119861

119887119878 [1198623

119860

119886119878 [1199101minus 1199102minus 1199103]]

(12)




119910out

=119860

119886(119901 + 119862

2119878 [1198621

119860

119886119878 [119910out]])

minus119861

1198871198624119878 [1198623

119860

119886119878 [119910out]]

minus119866

1198921198627119878 [1198625

119860

119886119878 [119910out] minus 1198626

119861

119887119878 [1198623

119860

119886119878 [119910out]]]

(13)



Scientifica 5



5= 1199106= 1199107= 1199108=



1199101

1199102

1199103

1199104

1 45Stable ndash0124 0008 6097 5882 0339 0174


2 38Stable 1018 0014 7037 5600 0419 0166


3 37 Unstable 5466 0106 31254 25750 0037 07634 8 Stable 10004 0226 31500 19258 2238 0571




1205821


1205822


1205823


1205824


1205825

minus3524 + 245119894 487 159 minus 788119894

1205826


1205827


1205828


1205829





1205821


1205822


1205823


1205824


1205825


1205826


1205827


1205828


1205829





Phase 3119861 = 37

Phase 4119861 = 8





1205824 minus1579 + 919119894 minus3522 + 236119894



1205827 207 + 902119894 minus995 minus 771119894


1205829 minus3516 + 219119894 minus995 + 771119894





6 Scientifica

05 10Time (sec)

minus05

0

05

Out

put (

mV

)

(a)

minus05

0

05

Out

put (

mV

)05 10

Time (sec)

(b)

minus05

0

05

Out

put (

mV

)

05 10Time (sec)

(c)









Scientifica 7

minus05

0

05

Out

put (

mV

)

05 10Time (sec)

(a)

minus05

0

05

Out

put (

mV

)05 10

Time (sec)

(b)

minus05

0

05

Out

put (

mV

)

05 10Time (sec)

(c)


Pyramidal cells

EPSP

EPSP

C1C2

C3C4

p 997888rarr

p 997888rarr

997888rarr p

997888rarr p

(a)

EPSPPyramidal cells

C1C2

C3C4

p 997888rarr

997888rarr p

997888rarr p

(b)




2 and the IPSP




4)


9have been





as follows

0(119905) = 119910

4 (14)

1(119905) = 119910

5 (15)

2(119905) = 119910

6 (16)

3(119905) = 119910

7 (17)

4(119905) = 119860119886119878 [119910

1(119905) minus 119862

41199102(119905) minus 119910

3(119905)] minus 2119886119910

4(119905)

minus 11988621199100(119905)

(18)

5(119905) = radic0001119860119886120590

2119903 (119905)

+ (119860119886 (119898 + 1198622119878 [11986211199100(119905)]) minus 2119886119910

5(119905) minus 119886

21199101(119905))

(19)

6(119905) = 119861119887119878 [119862

31199100(119905)] minus 2119887119910

6(119905) minus 119887

21199102(119905) (20)

7(119905) = 119866119892119862

7119878 [11986251199100(119905) minus 119862

61199104(119905)] minus 2119892119910

7(119905)

minus 11989221199103(119905)

(21)


8 Scientifica

EPSP

IPSP

IPSP

EPSP

C1C2

C3C4

C5

C6

C7

r(t) y1

y2

y3

y0

+

+

++

minus

minusminus


p 997888rarr

p 997888rarr

p 997888rarr

p 997888rarr

997888rarr p

997888rarr p

997888rarr p

997888rarr p





4 Discussion








Scientifica 9


Additional Points


Competing Interests


Acknowledgments


References


























10 Scientifica




























Neural Plasticity










Psychiatry Journal









Journal of


4 Scientifica


Phase12345

4538378

11

202020202

B G

B = 38

B = 8

B = 45

minus2

0

2

4

6

8

10

12

14

Out

put (

mV

)

5 10 15 20 25 30 35 40 450Time (sec)

minus2

0

2

4

6

8

10

12

14

you

t(m

V)






1199100=119860

119886119878 [1199101minus 1199102minus 1199103]

1199101=119860

119886(119901 + 119862

2119878 [11986211199100]) =

119860

119886(119901

+ 1198622119878 [1198621

119860

119886119878 [1199101minus 1199102minus 1199103]])

1199102=119861

1198871198624119878 [11986231199100] =

119861

1198871198624119878 [1198623

119860

119886119878 [1199101minus 1199102minus 1199103]]

1199103=119866

1198921198627119878 [11986251199100minus 11986261199104] =

119866

119892

sdot 1198627119878 [1198625

119860

119886119878 [1199101minus 1199102minus 1199103]

minus 1198626

119861

119887119878 [1198623

119860

119886119878 [1199101minus 1199102minus 1199103]]]

1199104=119861

119887119878 [11986231199100] =

119861

119887119878 [1198623

119860

119886119878 [1199101minus 1199102minus 1199103]]

(12)




119910out

=119860

119886(119901 + 119862

2119878 [1198621

119860

119886119878 [119910out]])

minus119861

1198871198624119878 [1198623

119860

119886119878 [119910out]]

minus119866

1198921198627119878 [1198625

119860

119886119878 [119910out] minus 1198626

119861

119887119878 [1198623

119860

119886119878 [119910out]]]

(13)



Scientifica 5



5= 1199106= 1199107= 1199108=



1199101

1199102

1199103

1199104

1 45Stable ndash0124 0008 6097 5882 0339 0174


2 38Stable 1018 0014 7037 5600 0419 0166


3 37 Unstable 5466 0106 31254 25750 0037 07634 8 Stable 10004 0226 31500 19258 2238 0571




1205821


1205822


1205823


1205824


1205825

minus3524 + 245119894 487 159 minus 788119894

1205826


1205827


1205828


1205829





1205821


1205822


1205823


1205824


1205825


1205826


1205827


1205828


1205829





Phase 3119861 = 37

Phase 4119861 = 8





1205824 minus1579 + 919119894 minus3522 + 236119894



1205827 207 + 902119894 minus995 minus 771119894


1205829 minus3516 + 219119894 minus995 + 771119894





6 Scientifica

05 10Time (sec)

minus05

0

05

Out

put (

mV

)

(a)

minus05

0

05

Out

put (

mV

)05 10

Time (sec)

(b)

minus05

0

05

Out

put (

mV

)

05 10Time (sec)

(c)









Scientifica 7

minus05

0

05

Out

put (

mV

)

05 10Time (sec)

(a)

minus05

0

05

Out

put (

mV

)05 10

Time (sec)

(b)

minus05

0

05

Out

put (

mV

)

05 10Time (sec)

(c)


Pyramidal cells

EPSP

EPSP

C1C2

C3C4

p 997888rarr

p 997888rarr

997888rarr p

997888rarr p

(a)

EPSPPyramidal cells

C1C2

C3C4

p 997888rarr

997888rarr p

997888rarr p

(b)




2 and the IPSP




4)


9have been





as follows

0(119905) = 119910

4 (14)

1(119905) = 119910

5 (15)

2(119905) = 119910

6 (16)

3(119905) = 119910

7 (17)

4(119905) = 119860119886119878 [119910

1(119905) minus 119862

41199102(119905) minus 119910

3(119905)] minus 2119886119910

4(119905)

minus 11988621199100(119905)

(18)

5(119905) = radic0001119860119886120590

2119903 (119905)

+ (119860119886 (119898 + 1198622119878 [11986211199100(119905)]) minus 2119886119910

5(119905) minus 119886

21199101(119905))

(19)

6(119905) = 119861119887119878 [119862

31199100(119905)] minus 2119887119910

6(119905) minus 119887

21199102(119905) (20)

7(119905) = 119866119892119862

7119878 [11986251199100(119905) minus 119862

61199104(119905)] minus 2119892119910

7(119905)

minus 11989221199103(119905)

(21)


8 Scientifica

EPSP

IPSP

IPSP

EPSP

C1C2

C3C4

C5

C6

C7

r(t) y1

y2

y3

y0

+

+

++

minus

minusminus


p 997888rarr

p 997888rarr

p 997888rarr

p 997888rarr

997888rarr p

997888rarr p

997888rarr p

997888rarr p





4 Discussion








Scientifica 9


Additional Points


Competing Interests


Acknowledgments


References


























10 Scientifica




























Neural Plasticity










Psychiatry Journal









Journal of


Scientifica 5



5= 1199106= 1199107= 1199108=



1199101

1199102

1199103

1199104

1 45Stable ndash0124 0008 6097 5882 0339 0174


2 38Stable 1018 0014 7037 5600 0419 0166


3 37 Unstable 5466 0106 31254 25750 0037 07634 8 Stable 10004 0226 31500 19258 2238 0571




1205821


1205822


1205823


1205824


1205825

minus3524 + 245119894 487 159 minus 788119894

1205826


1205827


1205828


1205829





1205821


1205822


1205823


1205824


1205825


1205826


1205827


1205828


1205829





Phase 3119861 = 37

Phase 4119861 = 8





1205824 minus1579 + 919119894 minus3522 + 236119894



1205827 207 + 902119894 minus995 minus 771119894


1205829 minus3516 + 219119894 minus995 + 771119894





6 Scientifica

05 10Time (sec)

minus05

0

05

Out

put (

mV

)

(a)

minus05

0

05

Out

put (

mV

)05 10

Time (sec)

(b)

minus05

0

05

Out

put (

mV

)

05 10Time (sec)

(c)









Scientifica 7

minus05

0

05

Out

put (

mV

)

05 10Time (sec)

(a)

minus05

0

05

Out

put (

mV

)05 10

Time (sec)

(b)

minus05

0

05

Out

put (

mV

)

05 10Time (sec)

(c)


Pyramidal cells

EPSP

EPSP

C1C2

C3C4

p 997888rarr

p 997888rarr

997888rarr p

997888rarr p

(a)

EPSPPyramidal cells

C1C2

C3C4

p 997888rarr

997888rarr p

997888rarr p

(b)




2 and the IPSP




4)


9have been





as follows

0(119905) = 119910

4 (14)

1(119905) = 119910

5 (15)

2(119905) = 119910

6 (16)

3(119905) = 119910

7 (17)

4(119905) = 119860119886119878 [119910

1(119905) minus 119862

41199102(119905) minus 119910

3(119905)] minus 2119886119910

4(119905)

minus 11988621199100(119905)

(18)

5(119905) = radic0001119860119886120590

2119903 (119905)

+ (119860119886 (119898 + 1198622119878 [11986211199100(119905)]) minus 2119886119910

5(119905) minus 119886

21199101(119905))

(19)

6(119905) = 119861119887119878 [119862

31199100(119905)] minus 2119887119910

6(119905) minus 119887

21199102(119905) (20)

7(119905) = 119866119892119862

7119878 [11986251199100(119905) minus 119862

61199104(119905)] minus 2119892119910

7(119905)

minus 11989221199103(119905)

(21)


8 Scientifica

EPSP

IPSP

IPSP

EPSP

C1C2

C3C4

C5

C6

C7

r(t) y1

y2

y3

y0

+

+

++

minus

minusminus


p 997888rarr

p 997888rarr

p 997888rarr

p 997888rarr

997888rarr p

997888rarr p

997888rarr p

997888rarr p





4 Discussion








Scientifica 9


Additional Points


Competing Interests


Acknowledgments


References


























10 Scientifica




























Neural Plasticity










Psychiatry Journal









Journal of


6 Scientifica

05 10Time (sec)

minus05

0

05

Out

put (

mV

)

(a)

minus05

0

05

Out

put (

mV

)05 10

Time (sec)

(b)

minus05

0

05

Out

put (

mV

)

05 10Time (sec)

(c)









Scientifica 7

minus05

0

05

Out

put (

mV

)

05 10Time (sec)

(a)

minus05

0

05

Out

put (

mV

)05 10

Time (sec)

(b)

minus05

0

05

Out

put (

mV

)

05 10Time (sec)

(c)


Pyramidal cells

EPSP

EPSP

C1C2

C3C4

p 997888rarr

p 997888rarr

997888rarr p

997888rarr p

(a)

EPSPPyramidal cells

C1C2

C3C4

p 997888rarr

997888rarr p

997888rarr p

(b)




2 and the IPSP




4)


9have been





as follows

0(119905) = 119910

4 (14)

1(119905) = 119910

5 (15)

2(119905) = 119910

6 (16)

3(119905) = 119910

7 (17)

4(119905) = 119860119886119878 [119910

1(119905) minus 119862

41199102(119905) minus 119910

3(119905)] minus 2119886119910

4(119905)

minus 11988621199100(119905)

(18)

5(119905) = radic0001119860119886120590

2119903 (119905)

+ (119860119886 (119898 + 1198622119878 [11986211199100(119905)]) minus 2119886119910

5(119905) minus 119886

21199101(119905))

(19)

6(119905) = 119861119887119878 [119862

31199100(119905)] minus 2119887119910

6(119905) minus 119887

21199102(119905) (20)

7(119905) = 119866119892119862

7119878 [11986251199100(119905) minus 119862

61199104(119905)] minus 2119892119910

7(119905)

minus 11989221199103(119905)

(21)


8 Scientifica

EPSP

IPSP

IPSP

EPSP

C1C2

C3C4

C5

C6

C7

r(t) y1

y2

y3

y0

+

+

++

minus

minusminus


p 997888rarr

p 997888rarr

p 997888rarr

p 997888rarr

997888rarr p

997888rarr p

997888rarr p

997888rarr p





4 Discussion








Scientifica 9


Additional Points


Competing Interests


Acknowledgments


References


























10 Scientifica




























Neural Plasticity










Psychiatry Journal









Journal of


Scientifica 7

minus05

0

05

Out

put (

mV

)

05 10Time (sec)

(a)

minus05

0

05

Out

put (

mV

)05 10

Time (sec)

(b)

minus05

0

05

Out

put (

mV

)

05 10Time (sec)

(c)


Pyramidal cells

EPSP

EPSP

C1C2

C3C4

p 997888rarr

p 997888rarr

997888rarr p

997888rarr p

(a)

EPSPPyramidal cells

C1C2

C3C4

p 997888rarr

997888rarr p

997888rarr p

(b)




2 and the IPSP




4)


9have been





as follows

0(119905) = 119910

4 (14)

1(119905) = 119910

5 (15)

2(119905) = 119910

6 (16)

3(119905) = 119910

7 (17)

4(119905) = 119860119886119878 [119910

1(119905) minus 119862

41199102(119905) minus 119910

3(119905)] minus 2119886119910

4(119905)

minus 11988621199100(119905)

(18)

5(119905) = radic0001119860119886120590

2119903 (119905)

+ (119860119886 (119898 + 1198622119878 [11986211199100(119905)]) minus 2119886119910

5(119905) minus 119886

21199101(119905))

(19)

6(119905) = 119861119887119878 [119862

31199100(119905)] minus 2119887119910

6(119905) minus 119887

21199102(119905) (20)

7(119905) = 119866119892119862

7119878 [11986251199100(119905) minus 119862

61199104(119905)] minus 2119892119910

7(119905)

minus 11989221199103(119905)

(21)


8 Scientifica

EPSP

IPSP

IPSP

EPSP

C1C2

C3C4

C5

C6

C7

r(t) y1

y2

y3

y0

+

+

++

minus

minusminus


p 997888rarr

p 997888rarr

p 997888rarr

p 997888rarr

997888rarr p

997888rarr p

997888rarr p

997888rarr p





4 Discussion








Scientifica 9


Additional Points


Competing Interests


Acknowledgments


References


























10 Scientifica




























Neural Plasticity










Psychiatry Journal









Journal of


8 Scientifica

EPSP

IPSP

IPSP

EPSP

C1C2

C3C4

C5

C6

C7

r(t) y1

y2

y3

y0

+

+

++

minus

minusminus


p 997888rarr

p 997888rarr

p 997888rarr

p 997888rarr

997888rarr p

997888rarr p

997888rarr p

997888rarr p





4 Discussion








Scientifica 9


Additional Points


Competing Interests


Acknowledgments


References


























10 Scientifica




























Neural Plasticity










Psychiatry Journal









Journal of


Scientifica 9


Additional Points


Competing Interests


Acknowledgments


References


























10 Scientifica




























Neural Plasticity










Psychiatry Journal









Journal of


10 Scientifica




























Neural Plasticity










Psychiatry Journal









Journal of














Neural Plasticity










Psychiatry Journal









Journal of


Documents

Research Article Analysis and Enhancements of a Prolific … · 2019. 7. 30. · of epilepsy that simulate the electroencephalogram (EEG) at the level of a neuronal population [ ]