A quantitative population model of whisker barrels: Re ... - CiteSeerX

Journal of Computational Neuroscience 3,247-264 (1996) © 1996 Kluwer Academic Publishers. Manufactured in The Netherlands.

A Quantitative Population Model of Whisker Barrels: Re-Examining the Wilson-Cowan Equations DAVID J. PINTO

Department of Mathematics/Statistics and Centerfor Neuroscience, University of Pittsburgh, Pittsburgh, PA 15261 pinto @whiskers.neurobio.pitt.edu

JOSHUA C. BRUMBERG AND DANIEL J. SIMONS

Department of Neurobiology, University of Pittsburgh, Pittsburgh, PA 15261 josh @whiskers.neurobio.pitt.edu simons @whiskers.neurobio.pitt.edu

G. BARD ERMENTROUT

Department of Mathematics~Statistics, University of Pittsburgh, Pittsburgh, PA 15261 bard @popeye.math.pitt .edu

Received November 1, 1995; Accepted February 28, 1996 Action Editor: Roger Traub

Abstract. Beginning from a biologically based integrate and fire model of a rat whisker barrel, we employ semirigorous techniques to reduce the system to a simple set of equations, similar to the Wilson-Cowan equations, while retaining the ability for both qualitative and quantitative comparisons with the biological system. This is made possible through the clarification of three distinct measures of population activity: voltage, firing rate, and a new term called synaptic drive. The model is activated by prerecorded neural activity obtained from thalamic "barreloid" neurons in response to whisker stimuli. Output is produced in the form of population PSTHs, one each corresponding to activity of spiny (excitatory) and smooth (inhibitory) barrel neurons, which is quantitatively comparable to PSTHs from electrophysiologically studied regular-spike and fast-spike neurons. Through further analysis, the model yields novel physiological predictions not readily apparent from the full model or from experimental studies. Keywords:

somatosensory cortex, vibrissae, cortical circuits, local networks, computer simulation

Abbreviations

1.

PW: AW: RSU: FSU: TCU: VPM: PSTH: PSP: CTR:

Strategies for modeling neuronal populations typically fall into one of two seemingly diametric categories. Realistic models strive to capture mechanisms involved in neuronal phenomena through the use of large sets of equations and parameters that incorporate as many of the known biological features as possible. This approach has the benefit of being closely related to the biological system, making quantitative comparisons and experimental predictions a natural by-product of the

Principal Whisker, Adjacent Whisker, Regular-Spike Unit (excitatory/spiny), Fast-Spike Unit (inhibitory/smooth), Thalamocortical Unit, Ventral Posterior Medical Nucleus, Peristimulus Histogram, Post-Synaptic Potential, Conditioned-Test Ratio.

Introduction

248

Pinto et aL

modeling process (e.g., Wilson and Bower, 1992). Abstract models, on the other hand, attempt to cull from the system all but its most essential elements in order to arrive at a small, relatively simple set of equations that capture the essence of the process. The benefit here is the level of clarity that comes from clearing away the details involved in a particular biological system. Also, the resultant set of equations is usually amenable to mathematical treatment, enabling the application of a full range of analytic techniques (e.g., Ermentront, 1995). In this article, we bring together beneficial aspects from each of these strategies. Beginning with a largescale, biologically based, integrate and fire model of a cortical neuronal network (Kyriazi and Simons, 1993), we apply semirigorous reduction procedures in order to construct an abstract model of the same System. Sufficient biological realism is retained so that the reduced model functions nearly as well as the large model from which it is derived, allowing both qualitative and quantitative comparisons with biological data. Using analytic techniques, the reduced model provides further insights into the system's function and yields novel biological predictions not readily apparent from either the full model or experimental studies. Rich in both function (Simons, 1995) and anatomical structure (Woolsey and Van der Loos, 1970), the whisker-barrel system of the rat primary somatosensory cortex provides an ideal environment for modeling studies geared toward understanding sensory processing. Barrels, neuronal clusters located in the granular layer of the rat somatosensory cortex, relate in a one to one fashion to individual whiskers on the contralateral mystacial pad (Welker, 1976). Individual neurons within a barrel have the shared feature of responding most robustly to deflection of the same particular whisker, the principal whisker (PW) for that barrel. A major source of afferent input to the barrel is the thalamic ventral posterior medial nucleus (VPM). Neurons in VPM are grouped into anatomical regions called "barreloids" which are also in one-to-one correspondence with the whiskers of the face (Land et al., 1995). The neurons of a barrel are linked into a local network and consist of two primary cell types, spiny stellate cells and smooth cells. Spiny cells are the most predominant and are thought to comprise the excitatory element of the network. Smooth cells are less common and are considered to be inhibitory in nature (McCormick et al., 1985; White, 1989). Both types can be distinguished according to features of

their respective wave-forms acquired via extracellular recordings. Spiny cells are thought to discharge "regular-spikes", and will be equivalently referred to as regular-spike units (RSUs), whereas smooth cells have been shown to discharge fast spikes, and will be described as fast-spike units (FSUs) (McCormick et al., 1985; Simons, 1978). A typical stimulation paradigm for the activation of either thalamic or cortical neurons consists of an initial rapid deflection of the PW, holding it in a deflected state for a period of time, and returning the whisker to its original position. The neuronal response to such stimulation can be subdivided into three distinct phases; the ON response, which lasts for approximately 20 msec following the initial deflection; the Plateau phase, occuring during the 200 msec period of maintained whisker deflection; and the OFF response, lasting approximately 20 msec following the whisker's return to its original state. Whiskers are deflected singly or in paired combinations of the PW and an adjacent whisker (AW). Spike data from multiple whisker deflections are gathered into peristimulus time histograms (PSTHs) and reflect the average neuronal response to a whisker stimulus or, when a PSTH includes responses from many neurons, the average population response (see, e.g., Fig. 3). Simons and Carvell (1989) compared the response properties of FSUs and RSUs in barrels and thalamocortical units (TCUs) in barreloids. The FSU population was found to be the most responsive to stimuli, having high levels of spontaneous activity and discharging vigorously to deflection of both PW and nearby surrounding whiskers. By contrast, RSUs are less spontaneously active, have smaller excitatory receptive fields, often responding only to deflection of the PW, and display robust levels of surround or "crosswhisker" inhibition (see also Simons, 1985). Of the three cell types, RSUs display the greatest response differential between strong versus weak input, such as the response to initial whisker deflection versus whisker return or deflection of the PW versus AW. TCUs, like FSUs and unlike RSUs, have multiwhisker receptive fields, display only weak levels of surround inhibition (Nicolelis et al., 1993; Sugitani et al., 1990) and show smaller differentials to strong versus weak stimuli (Brumberg et al., 1995; Simons and Carvell, 1989). These cortical response properties necessarily involve a spatiotemporal transformation of the thalamic input signal. In the case of RSUs in particular, the net result of this transformation is an increase in the

A Quantitative Population Model of Whisker Barrels

signal-to-noise ratio, an enhancement of signal contrast in both the spatial and temporal domains, and the production of a baseline signal for further processing within the cortical column. The transformation itself has been hypothesized to emerge from four principle features of barrel organization: nonlinear neuronal properties, differential responsiveness of the RSUs and FSUs, convergence of thalamic input onto each population, and interconnections among and between RSUs and FSUs within the barrel (Simons and Carvell, 1989). The goal of previous modeling work (Kyriazi and Simons, 1993) was to evaluate this hypothesis by incorporating these principles in a computational framework and to illustrate their ability to capture the net effects of spatial and temporal integration within a barrel. By contrast, the goal of the present study is to reduce this computational framework as much as possible, while retaining the features essential for meaningful comparisons with the biological system. 2.

Methods

2.1. Summary of the Full Model The model barrel of Kyriazi and Simons consists of 100 barrel neurons, 70 excitatory, 30 inhibitory. Each unit receives synapses, both excitator3~ and inhibitory, from other units in the network as well as from thalamic input neurons (described below). For a given unit, the number and strength of the synapses projecting to that unit are randomly distributed about a prespecifled mean value set for each of the two populations. The ratio of excitatory to inhibitory units, the mean number of synapses made by units of each population, and the relative strength of the synapses within each population are based on estimates from published light and electron microscopy studies (White, 1989; Keller, 1995). For instance, the mean strength of thalamocortical synapses onto inhibitory units is set to be greater than those onto excitatory units. This reflects the anatomical finding that thalamocortical synapses tend to occur at proximal locations on smooth (inhibitory) cells, whereas on spiny (excitatory) cells they occur almost exclusively on spines distributed along the length of their dendrites (Rall, 1967; White, 1989). Each barrel neuron is modeled by an equation describing its membrane potential and incorporates both spatial and temporal synaptic integration. The membrane potential is computed by summing synaptic inputs from each population as well as decayed values

249

from previous synaptic events. This value is passed through an activation function (different for each population) to determine the probability of the unit firing in that particular time step. For each time step, this probability is used to determine whether or not a unit actually produces an action potential. Action potentials produce synaptic events, or postsynaptic potentials (PSPs), in postsynaptic units. In the model, PSPs rise instantaneously and decay exponentially, with a different time course for excitatory and inhibitory synapses. When a unit produces an action potential, a refractory period is imposed so that the unit is prevented from producing further action potentials for a set number of time steps. The model's equations and optimal parameter values are summarized in Appendix A. The full model has proven effective in reproducing many of the operational features of a real barrel under several different stimulation protocols (Kyriazi and Simons, 1993; Kyriazi et al., 1996). A question that arises, however, is whether the full complexity of the system is necessary for capturing the appropriate response properties. A reduced model, which incorporates the four features of barrel organization mentioned above, and little else, and which is also able to capture the same operational features as the full model, would provide strong support for the contention that these features are indeed sufficient for explaining the observed cortical responses. Such a model would also allow for further insights to be gained by means of analytic techniques unavailable in the case of the full model and could be readily extended to encompass larger regions of barrel cortex, including columnar structure and multiple barrel domains, without greatly increasing computational overhead. 2.2. The Reduced Model Details of the reduction from the full to reduced model are presented in Appendix B. The equations and typical parameter values are summarized in Appendix C. During the investigation of the reduction process, it was realized that the emerging equations were nearly identical to the Wilson-Cowan equations (Wilson and Cowan, 1972). This was particularly interesting in that, even though the Wilson-Cowan equations have been extremely successful in providing qualitative descriptions of several neural processes, we have been unable to locate any instance in which they have also been employed for quantitative comparisons.

250

Pinto et al.

Since the reduced model derives from a system that is capable of quantitative comparisons, compensating for the various assumptions and approximations introduced in the reduction process should be sufficient to allow for just such analysis. The remainder of this section describes our attempts in this direction.

2.3.

Spike Train Data as Model Input

A unique and beneficial feature of the original model is the use of real biological data, in the form of prerecorded spike-trains from individual thalamic barreloid neurons, as external input. In this fashion, model cortical neurons can be made to respond to the same thalamic activity seen by real cortical neurons during typical whisker stimulation protocols. This is the key element of the full model which allows for direct and quantitative comparisons to be made between the model's output and biological data. The same advantage can be gained in the reduced model with proper treatment of the thalamic input data. Rather than individual spike trains, the reduced model requires some form of population response as input. This is obtained by compiling the spike data into peri-stimulus time histograms (PSTHs), which, after proper scaling, may be interpreted as a function describing the average firing rate of the thalamic neuronal population over time. Determining the proper way to insert these data into the reduced model requires a closer examination of the equations themselves. In final form, the reduced equations succinctly define the relationship between three distinct measures of neuronal activity: voltage, firing rate, and a measure which we describe as synaptic drive (discussed below) (Fig. 1). In order to incorporate thalamic data into the simulation, it must be first transformed into a representation of activity in terms of synaptic drive, S r (t). Since the PSTH essentially reflects the firing rate of the thalamic cells, the following

equation must be introduced in order to effect this transformation: dS r re---~- + ST(t) = PSTH(t). The same observation indicates which aspect of the model's output will be directly comparable with biological results. While the solution to each equation provides a measure of synaptic drive in the population, the data to which it is compared has the form of population firing rates (i.e., population PSTHs from RSUs and FSUs). Thus it is an auxiliary term, the right side of the equation in Fig. 1, which provides the means by which to make direct and quantitative comparisons with the biological system.

2.4.

Tuning the Model

The transformation from the full to reduced model requires several assumptions and approximations (see Appendix B). Those that have the greatest impact on the model's function are the lack of a refractory period and the linear manipulation of the nonlinear activation function. While it is not possible, or at least extremely difficult, to alleviate these sources of error in a strictly analytic fashion, a careful examination of the system and its function suggests means by which at least partial compensation may be possible (for analytic treatments, see Ingber, 1982 and Ginzberg and Sompolinsky, 1994).

Activation Function. Simulation results indicate the step in the reduction process that introduces the greatest amount of error is the approximation of the average firing rate by the firing rate due to average voltage, that is, if-017° 2.j I Pe(V;) ~ Pe ( ~17j--~l Vf ) . Correcting for the difference between the two entails reshaping the nonlinear, single-unit activation function into a function that captures the voltage-to-firing rate relationship for the population.

Firing rate r

dS E

+

st(t)

= Re

Synoptic Drive

Figure 1. Relationshipbetweenthreemeasuresof activity.

/

\ g(t) +

tc g ( t ) Voltage

ic S1(t).l

7


1.0

0.9

Single Cell Population

0.8

Adj. Single Cell

f

]

0.7

rr r

0.60.50.4 0.30.20.1-

/

/

.f i .~.:..'/:J

0.0--45 -44 -43

-42 -41 -40 Voltage (mV)

-39 -38 -37

Figure 2. Adjustment of the activation function to account for population activity. For a small (n = 5) regularly distributed population (D) the average of firing rates obtained from individual voltages (Q)) is greater than the firing rate obtained from the average voltage (center D). Performing this calculation about mean voltages throughout the domain yields the population activation function, also seen to be sigmoidal. One means by which to compensate for this increase in variability is to adjust the slope of the single unit activation function. Note the discrepancy introduced by this approach in both the low and high voltage regimes.

A portion of the error involved can be best understood through a simple example. Consider a small group of neurons with voltage levels distributed regularly about some mean value near the lower cusp of a particular activation function (Fig. 2). Due to its nonlinear nature, especially near the cusp region, the firing rate obtained from the average voltage will be much lower than the average of the firing rates obtained from each voltage individually. The amount of discrepency between the two depends on where along the activation function the mean voltage level of the population occurs and the nature of the voltage distribution within the population. For a population with normally distributed voltage levels about any .given mean, the proper activation function remains sigmoidal, but with a smoother rise in the linear regime (i.e., the function is less deterministic). One way in which to compensate,

251

then, for the error introduced by the approximation is to align the slope of the single-neuron activation function to match that of the population. Simulation results show this method to be effective except for too high a level of spontaneous activity produced by the model, as can be seen in Figs, 3 and 6b. This is expected in that the range of the adjusted activation function reflecting low voltage levels lies well above the desired population curve. We attempted to compensate for this through the use of various sigmoid-like functions with differing degrees of curvature around the cusp regions, eventually settling on the error function as the sigmoid of choice (Appendix C). To achieve reasonable results, it was also necessary to apply a scaling term to the activation functions (Appendix C). The biophysical interpretation for this is unclear. One explanation is that the term acts as a conversion factor between firing rate (spikes/sec) and synaptic drive (1/synapse, see discussion), ensuring consistency in the units of the equation. This constitutes the primary difference between the present model and the Wilson-Cowan equations in that the nonlinear function is now capable of achieving values greater than one. Operationally, however, since the firing rates produced by the model are extremely low (see Fig. 3), this rarely occurs. Further, as the primary variable is no longer being interpreted as a probability function (see discussion and Appendix D), as it was in the original Wilson and Cowan paper, the bounds of the nonlinearity are not as crucial. Determining the exact parameter values necessary to adjust accurately for the averaging process requires more empirical data than is currently available. For instance, the proper slope for the activation functions entails knowledge of the mean voltage level for each neuronal population, and the distribution of those voltages (which is likely also to depend on mean voltage). An optimization routine was therefore utilized to find reasonable values for the slope and scale of the nonlinear functions for each of the two populations (discussed below). In the full model, the slope of the activation function was described in terms of its temperature, with a lower temperature corresponding to a steeper slope. Interestingly, the relationship between the temperature of the RSUs and FSUs differs between the full and reduced models. In the full model, the single-neuron activation function of the smooth cells was assigned a higher temperature than that of spiny cells, reflecting a greater degree of excitability in the former and a greater degree

252

Pinto et al.

.t

_

i

RSU/Exc

L

FSU/Inh

[

I 0.11., 20ms

TCU

Real

Simulated

Figure 3.

Population profile from standard whisker deflection protocol PSTH on the left shows biological spike data accumulated from 242 RSUs, 16 FSUs and 135 TCUs in response to 10 repetitions in each of eight directions of PW deflection (Simons and Carvell, 1989). The plots on the right show the firing rate output of the reduced model in response to input in the form of a TCU PSTH represented in the lower graph as described in the text. Vertical scale may be interpreted as the average firing rate of each population.

of nonlinearity in the latter. The reduced model, on the other hand, obtains best results when the smooth cell activation temperature is lower than that of the spiny cells. This can be explained by the fact that the averaging process for the spiny cell population includes many more cells than the smooth cell population, increasing the variability to a greater extent. This variance is expressed in the slope of the activation function along with the level of excitability for individual neurons. Taken together, the excitability and the variance result in a higher "population temperature" for the spiny cells as compared to the smooth. This difference between single cell and population activity is illustrated empirically in the population transfer functions obtained from single unit activity in the original model (Fig. 7; Kyriazi and Simons, 1993). Refractory Period. The original model imposed a refractory period on individual units such that, if a unit had fired within the past few msecs, another spike could not be produced regardless of the unit's voltage level. This strategy proved difficult to adapt into the formal reduction and so was originally neglected. Introducing refractory elements into the reduced equations is problematic in its own right. By definition, a refractory state depends on the prior activity within the population. However, only the present activity is available from within the reduced equations. And while prior activity could be tracked in an

ad-hoc fashion throughout the course of the simulation, this would preclude the application of mathematical analysis on the equations of the reduced model. Guided by the original model of Wilson and Cowan, a refractory element was introduced into the reduced equation as follows: dS e re ,,t----t'--I- se ( t ) : (1 -- e r f ( s e ) ) P e ( t O e e e c S

e -b . . . .

WieicSi),

Refraction

where er is a parameter loosely corresponding to the length of the refractory period, and F ( S e) is some function of S e of the form F ( S e) = S e, F ( S e) = Pe(Se), or d_~F = G ( s e ) , where G(S e) is some arbitrary function dt (several forms were tried). The first form for F (S e) corresponds to the refractory term used in the originalWilson-Cowan equations. The third was used in an attempt to account for the dynamic nature of refraction. However, the added complexity of two new differential equations (one each for the excitatory and inhibitory populations), as well as the various additional necessary parameters, did not seem justified in that the performance of the model was only slightly affected. The second form for F ( S e) was adopted as standard for the model to ensure that the refractory effects of activity remained bounded between zero and one. Also,


253

it seemed intuitively feasible that, in terms of the different activity measures discussed above, the firing rate is best suited to reflect the refractory elements present in the system. Since er and ir play the dual role of representing both the time lengths for each of the refractory periods as well the scale factor required for estimates of prior from present activity, we chose to use parameter values that differed from the original model. As in the case of the activation function, a parameter optimization scheme was utilized to determine appropriate values for the refractory parameters (discussed below). Surprisingly, by using different values for the activation and refractory parameters, the model produced reasonable results for all three forms of F(Se). In fact, if the refractory element was dropped entirely, values for the activation parameters alone were easily established which allowed the model to perform adequately. It seems that, in a population model, the influence of any type of refractory period can be accounted for simply by adjusting the shape of the nonlinear activation function. Similar observations were noted by Zetterberg etal. (1978) in an analysis of the Wilson-Cowan system as a model for EEG activity (see also Skondras, 1988).

measure (SCORE4) of accuracy in the reduced model. The remaining two features were neglected since they depend heavily on low level activities which, as described above, is a weak aspect of the reduced model. Using a global optimization algorithm based on the method of gradient-descent (Press et al., 1988), parameter values were determined that produced reasonable values for SCORE4. Once established using the two key features, the values remained unchanged throughout the various stimulation protocols and were used to produce the remaining results described below.

2.5.

The use of real biological input combined with a formal reduction from a more realistic model allows for both qualitative and quantitative comparisons between biological data and the reduced model's output. With the exception of the activation and refractory terms mentioned above, all of the results are based on parameters identical to those used in the original model. The remaining parameters were set to yield reasonable values of SCORE4. Figure 3 compares population PSTHs from real barrel neurons to the model's output in response to a single whisker (PW) deflection. The model clearly captures several features of the cortical response. For instance, the FSU population is considerably more active both with respect to stimulus evoked and low-level spontaneous activity, while the RSU population is less active and exhibits a greater differential between its response to the "strong" ON input as compared to the "weak" OFF input. The most prominent discrepency between the real and simulated data is the lack of temporal dispersion in the stimulus evoked responses. In particular, the model's responses rise more rapidly and are more temporally focused than those of the real populations in response to thalamic stimuli. As in the full model,

Optimization

One of the benefits gained from a stepwise reduction of the full to the reduced model is that most of the parameter values used in the original model (synaptic weights, convergence, decay constants, ans so on.) can be transfered directly to the reduced equations (Appendices A and C). The exceptions, as explained above, include the parameters involved in shaping the refractory periods and activation functions. In the original model, the accuracy of a proposed set of parameter values was tested using a measure called SCORE13. SCORE13 is an error term that depends on nine aspects of the full model's output as compared to desired values. Four of these nine aspects were deemed of particular importance in capturing the functionality of the barrel network and therefore weighed more heavily in the analysis. These included the ON response magnitude, the plateau response magnitude, plateau waveform matching measurements, and the ON response to OFF response ratio. In order to determine reasonable values for the remaining parameters in the reduced model, two of the four key features from SCORE 13, ON response magnitude and ON to OFF ratio, were used to construct a new

3.

Results

Simulations were run on an HP9000 series workstation. Simulation software designed specifically for the task was written in C and X Windows/Motif. The equations were solved numerically using a simple forward Euler algorithm with the time step of 1 msec governed by the temporal resolution of the thalamic PSTH input files. Phase plane analysis was conducted using the XPP software package (Ermentrout, 1994).

3.1.

Biological Comparisons

254

Pinto et al.

this can be partly accounted for in terms of the instantaneous rise time of simulated PSPs. The effect is further accentuated in the reduced model through the process of time-course graining and averaging as discussed in Appendix B. The difference becomes less pronounced if the total response activity is considered (that is, the areas beneath the response curves), in which case the model closely matches the biological response (see below). In order to assess the model's ability to reproduce quantitatively biological function, a series of simulations were conducted using various stimulation protocols. The Standard protocol, as described in the introduction, entails deflection of the PW alone. The measurements obtained from this protocol were used to derive reasonable activation and refractory parameter values in the reduced model. Once these parameters were set using the Standard protocol, the performance of the model was tested with the remaining protocols without further alteration. The Long Plateau protocol is similar to the Standard protocol except for the Plateau phase, which is extended from 200 msec to 1400 msec. This protocol is useful for the examination of the model's effectiveness in capturing temporal integration features of the system (Kyriazi et al., 1994). Inspection of the thalamic PSTH reveals that the thalamic OFF response is more temporally dispersed than the ON response under the Standard protocol (Figs. 3 and 6A). Using the Long Plateau protocol, however, the spikes in the OFF response generally occur earlier, creating a distribution similar to that of the ON response (Fig. 6A). The ratio of ON response to OFF response (ON-toOFF ratio) is an indicator of the extent to which this distribution effects the response levels in the cortical populations. As presented in Fig. 4, the high ON-to-OFF ratio of the RSU population using the standard protocol illustrates the sensitivity of the population to temporal properties of external influences. With the Long Plateau protocol, the ratio is closer to unity, reflecting the similarity between temporal distributions of thalamic ON and OFF inputs. This difference in biological response is also illustrated qualitatively in Fig. 6A. Features of spatial integration are investigated using the Cross Whisker protocol. Consisting of the deflection of an adjacent whisker (AW) 20 msec prior to PW deflection (followed by overlapping Plateau phases and staggered return), the Cross Whisker protocol allows insight into receptive field properties and surround inhibition.

Standard

Long Plateau

2.25

[] RSU/Exc

~©

I

1.5

i ;/

0.75 I j

.,...

0 ' VPM Barrel Model VPM Barrel Model Figure 4. Temporal integration. The left panel showsdata for the Standard protocolin which a 200 msec plateau period was used. In the fight panel, the plateau periodlasted 1400 msec. The high ON/OFF ratio of the RSU population reflects the sensitivity of that populationto differencesin temporaldistributions of external influence.

Examining the ON response for both PW and AW deflection reveals the excitatory receptive field characteristics of the two cortical populations (Fig. 5A). Despite the similarity in magnitude between the TCU and RSU response to PW deflection, the AW response of the RSU population is dramatically decreased in comparison with the TCU. This illustrates the tight focus of the RSU receptive field, as well as the differential responsiveness of RSUs to strong versus weak input. The FSUs, by contrast, respond robustly to both PW and AW deflection, exhibiting a broad receptive field. Properties of surround inhibition may be studied by comparing the ON response induced by PW stimulation "conditioned" with prior deflection of the AW, to the ON response when the PW is deflected alone. The ratio of these two responses is called the conditionedtest ratio (CTR) and provides a quantitative measure of surround inhibition. CTR values less than one indicate that the conditioning stimulus (AW) has inhibited the test stimulus (PW) whereas values greater than one indicate facilitation. Figure 5B compares the inhibitory effects of AW conditioning between the RSU and TCU populations. The thalamic response can account for only about 20% of the reduction seen in the RSU response, with the remainder dependent on inhibition and integration within the barrel network. To illustrate the dependence of this phenomena on network integration, the simulation was run both with and without corticocortical connections. With all intracortical population weights set to zero,


A)

Principal W h i s k e r Adjacent W h i s k e r

2.5

~2.0

[ ] RSU/Exc •

FSU/Inh

a) r.~

~1.0

O'~, r;?

.5

VPM B a r r e l Model VPM Barrel Model

B)

Cross-Whisker Inhibition

O

o~.~

o~

1.0-

~.

RSU/Exe

.._

0.&

:!~iiiiill

o£r o 0.4-

• ~'-'I

"~ 0.2o 0.0

1

VPM B a r r e l

Real

~'}!!!

+Net

-Net

Mode/

Figure 5. Spatial integration. A) ON response data for both the PW and AW deflected alone. Note the wide spatial receptive field apparent in the FSU populaton as compared to RSUs. B) Quantification of the effect of cross-whisker inhibition on the RSU population. The effect vanishes in the absence of corticocortical network connections, implicating network processes as the source.

the CTR ratio of the RSUs becomes a simple reflection of the thalamic response.

3.2. Temporal Integration The temporal profile of the thalamic population response varies considerably from one stimulus condition to the next. For instance, the thalamic PSTH corresponding to the ON response exhibits a sharp rise in activity followed by a steady decay. The OFF response produced by the Long Plateau protocol is similar. However, the OFF response of the Standard protocol shows some delay to peak activation as well as a slightly lower peak amplitude. These profiles, as well

255

as the corresponding RSU population responses, are presented in Fig, 6A. To explore more fully the effects of input distribution on cortical response, the reduced model was presented with artificial "population" response functions which approximate the distribution of thalamic activity under different conditions. Figure 6B shows the response of the simulated RSU population to three such "input triangles" in which it can be seen that the onset velocity of thalamic input has a profound effect on the level of the RSU response. Except for an initial sharp rise in the case of instantaneous onset velocity, the FSU population response mimics the triangular shape of the TCU input (data not shown). The difference in peak amplitude between the ON and OFF response of the Standard protocol raises the issue of whether stimulus onset velocity, amplitude, or perhaps duration is the key factor that determines the level of cortical response. This was addressed using the reduced model by systematically varying the input triangles in these three dimensions, with results presented in Figs. 6C and D. RSU responsiveness is clearly seen to be strongly dependent on stimulus onset velocity as compared to either duration or amplitude. Only under the nonphysiological condition of instantaneous rise time does amplitude play a significant role. This response to rapid changes in thalamic input can be explained in terms of network interactions. Because of the strong thalamic influence on the FSU population, the network is rapidly overwhelmed by inhibition. However, the time between the arrival ofthalamic activity at the cortex and the full effect of inhibitory mechanisms in the network provides a window of opportunity for the excitatory response. Its overall magnitude will depend on the amount of activity that reaches the RSU population before the window is forced closed by the wave of inhibition. Note, for instance, the complete cessation of activity following both the ON and OFF responses in Figs. 3 and 6. The differential responsiveness of the RSU population to onset velocity is consistent with a role of contrast enhancement, in this case in the temporal domain. Along with the above analysis, this suggests an easily tested experimental prediction. Assuming a thalamic response with temporal properties consistent with whisker motion (also easily tested), the model predicts that the RSU population will be preferentially stimulated by rapid whisker deflections as compared to those which are either large in amplitude or long in duration.

Pinto et al.

256

A)

i

Standard OFF

ON

Long Plateau OFF

1.6

C) ~

Amolitude

r~ _1.4

[] : .22

o

\\

- .20 A .18

O~.~

Base

C ~D

-1.2~

-

O

0...~ ------~-'2-5ms _1.0

J

I

I

0

0 . 0 ~T M

I

.

30 205

B)

Time (ms)

l

1

235 1405

_0.6~

|

1435

I

0.5~

Ak

I

I

I

Velocity

D) Simulated RSU

I

o25

0.4 ~'~

1,6

Velocitv

~o

[ ] - ¢~ O-2_2 -1.4 -4.4 A - 1 . 5 t~ a t - l . 1 -1.2 o

o

-1.0 i-do tip

0.0 ,

50ms

i

-.0.8 ~ .

o I

.22

o-----o I

.20

I

-0.6 ~r~ 0,4 " ~

.18

Amplitude R e s p o n s e to t e m p o r a l distribution o f input. A) Biological data illustrating thalamic responses to various modes of whisker deflection and the resultant response from RSUs (Kyriazi et al., 1994). B) Examples of the model's response to differences in the temporal distribution of input stimuli. C) and D) Systematically varying the base, amplitude, and onset slope of the input triangles reveals slope to be the determining factor governing the model's response. Only when onset is most steep does amplitude play a role in cortical responsiveness. Figure 6.

3.3.

Spatial Integration

In the original modeling paper the idea of tension as a network property was introduced. Tension is defined as the ratio of intrinsic to extrinsic excitatory influences impinging on a given cell population, where influence is measured as the product of synaptic weights and convergence, that is, Wee • e c

Tension -= - -

Wte " tc "

Expressed strictly in terms of excitatory synapses, tension acts to capture the relationship between recurrent and feedforward excitation (Douglas et al., 1995). In order to prevent runaway excitation, and to maintain a realistic level of excitability in the network, inhibitory strengths must also be altered as tension is varied. It was found that the best match to biophysical data was achieved when network and thalamic influences were approximately balanced (i.e., tension = 1).


This concept of tension can be incorporated into the reduced model in a straightforward manner:

257

A)

dS e

Te - - -- se(t) dt

+Pe ( ~ ( W e e ecSe-loieicSi)-[-WtetcSt), \Tension

where ~ = 1 corresponds to the baseline case in which external and internal excitatory influences are in balance. Changes in ~ from baseline reflect increases or decreases in the level of network tension. Analysis of the role tension plays in the reduced model relies on the relative simplicity of the equations involved. In this case, phase plane analysis of the twovariable system reveals that the steady-state condition may be characterized as a stable spiral. Perturbations to the system are invariably followed by a return to rest in the form of a damped oscillation. This corresponds to the empirical characterization of barrel activity as "cyclic alterations in unit excitability" (Simons and Carvell, 1989) and can be observed in the expanded population PSTH as an activation peak followed by a postexcitatory trough and secondary peaks. When tension is included in the phase plane analysis, the damped oscillations become faster and smaller in amplitude as tension is increased. Formally, the real portions of the complex eigenvalues become more negative. Intuitively, just as a tense spring snaps back to rest more quickly than a loose one when perturbed, so does a tense network snap back faster when perturbed by a thalamocortical volley. This is illustrated in Fig. 7A which shows the response of the model's activity under different levels of tension. As tension is increased, the postexcitatory trough becomes more narrow and less deep. It is of interest to note that this behavior is dependent on scaling of the activation function as mentioned above. The original Wilson-Cowan model, which lacked this scaling factor, is characterized by monotonic return to rest (Fig. 3; Wilson and Cowan, 1972). It has been shown (Simons and Carvell, 1989) that the level of cross-whisker inhibition depends on the time interval between adjacent and principle whisker deflection. The temporal relationship, in fact, corresponds to the time course of the return to rest oscillation, specifically the postexcitatory trough. Taken with the above discussion, it might be concluded that network tension level will influence both the amplitude

B)

.D..j~'O'O'n 0.8

.~'~'.I~I' o -v..O..o.~ o

.el

aS

~,)0.6

#s

.o

d

O'

i

D TCU

[~.-

a~ d

0.4

d

~"

0,2

I

I

I

I

I

I

l

1

1.2

1.4

1.6

1.8

2

[]

RSU

.... o ....

0.8

FSU

Tension

Figure 7. Effectof tensionon networkactivity. A) Effect of increased tension on postexcitatory suppression response following whisker deflection. The curly bracket indicates the region of whisker response magnified in the lower trace. The same effect is seen following the ON response, but is more pronounced in the case of the weaker OFF response or the AW response. B) Quantification of the predicted influence of changes in tension on surround inhibition for each population (CTR-conditioned-test ratio).

and time course of cross-whisker inhibition, measured in terms of the CTR. Figure 7B illustrates the predicted relationship between network tension and cross whisker inhibition. As tension in the model increases, the percent reduction in cortical response due to adjacent whisker deflection also decreases as expected. This finding is consistent with previous modeling studies (Kyriazi and Simons, 1993) that support the hypothesis of Simons and Carvell (1989) that intrabarrel processing is sufficient for explaining cross whisker inhibition, without the need for either multiple barrels or specific

258

Pinto et al.

membrane properties such as after-hyperpolarization (neither of which are included in the model).

4.

Discussion

The success of any reasonable modeling program should be measured both in terms of its ability to clarify complex biological phenomena as well as its capacity for generating relevant and testable predictions not readily apparent by other means. Beginning from a large-scale realistic model of a rat whisker barrel, we have employed semirigorous techniques to reduce the system to a simple set of equations, similar in form to the Wilson-Cowan equations, while retaining the potential for both qualitative and quantitative comparisons with the original biological system. The reduced model represents the minimal computational framework capable of capturing the dynamical features that we believe essential for understanding the effects of spatial and temporal integration within a barrel. Further clarity has been provided by defining a specific relationship between three distinct measures of population activity, voltage, firing rate, and synaptic drive. Finally, the model has yielded novel physiological predictions not readily apparent from the full model or experimental studies. In the temporal domain, simulation results indicate that the RSU population is most sensitive to rapid changes in thalamic activity as compared to either amplitude or duration. This leads to the prediction that rapid whisker deflection is the preferential stimulus for initiating RSU activity. This is contrary to a previous study in which the response of barrel neurons were concluded to be more dependent on stimulus amplitude (Ito, 1985). In order to test our prediction, as well as explore this discrepency more fully, we are currently conducting experiments in which the Standard protocol has been modified to include systematic variations in amplitude, duration, and velocity of deflection. It will also be necessary to determine thalamic responses under similar conditions to validate the assumption that temporal properties of whisker motion are reflected in the thalamic response. A better understanding of temporal integration in the barrel system may be useful in explaining the basis for certain behavioral strategies employed during discrimination tasks. Good performance on such tasks is characterized by a specific range of whisking force and frequency (Carvell and Simons, 1995). The benefits

derived from such behavior may be due to the temporal nature of the response generated in thalamic and subsequent cortical activity. In the spatial domain, the model identified the concept of tension as a key element governing the effects of surround inhibition on the excitatory response. By increasing tension in the model, suppression of PW response, induced by prior AW deflection, decreases. The resulting prediction is that alterations of tension in the network will influence the level of cross-whisker inhibition. Validating this prediction experimentally may prove challenging. The difficulty lies in determining a viable experimental means by which to alter network tension without effecting other functional aspects of the system. Application of either receptor agonists or antagonists would impact either the level of activity or sensitivity of the system (essentially shifting or scaling the activation function) but would have little effect on the underlying connectivity that tension describes. Furthermore, alterations that influence corticocortical and thalamocortical connections identically would also have little effect since tension describes a measure of the ratio between the two. One possible means by which to differentially influence the synaptic efficacy of the two types of excitatory connections is to change the temperature in the surrounding cortical environment (Moser et al., 1993). This is also problematic, however, in that changes in temperature are likely to result in numerous incidental effects. Other possibilities include selective but reversible lesions to a subset of thalamocortical projections or application of carbachol, which has been shown to have differential effects on intrinsic versus extrinsic excitatory connections in both hippocampus and piriform cortex. (Hasselmo and Schnell, 1995; Hasselmo and Bower, 1992). Overcoming these practical difficulties may prove worthwhile in that the concept of tension is likely to be useful in understanding many aspects of cortical function. In the model, a tension level that was either too high (Fig. 7B) or too low (Fig. 5B) resulted in degradation of network performance, indicating the need for a balance between intrinsic and extrinsic influences. Changes in synaptic efficacy, and hence changes in tension, may be a key mechanism for the incorporation of learned behavior in a network (Recanzone et al., 1992). As such, tension is positioned to play a pivotal role in bridging the gap between such morphological changes and network function.


Since their original description, the Wilson-Cowan equations have proven extremely useful in providing qualitative descriptions of neuronal behavior at various levels. Understanding of the interactions between excitatory and inhibitory elements at the level of single cells (Kleinfeld and Raccuia-Behling, 1990; Ermentrout and Kopell, 1991), small networks (Evans, 1989; Matsushita et al., 1995), large scale activity such as EEGs (Lopes da Silva et al., 1974; Lagerlund and Sharbrough, 1989), and even intersystem connections (Tolbert et al., 1976) has been dramatically improved by employing the equations to explain various dynamic phenomena such as bistability (Sejnowsky, 1976; Barnwell and Stafford, 1977; Klaasen and Troy, 1984), hysteresis (Anninos and Argyrakis, 1983; Hoffman, 1986), limit cycle oscillations (Kawahara, 1980; Paydarfar et al., 1986; Ermentrout and Troy, 1989), and the general behavior of non-linear systems (Bertholon et al., 1988; Britton and Skevington, 1989). The barrier that has thus far constrained the system to purely qualitative descriptions, however, has been the lack of clarity in interpreting the biological significance of the variables involved (Wright et al., 1987; Lockery, 1993). In past work, the behavior of the system has been compared to firing rate (Cope and Tuckwell, 1979; Brannan and Boyce, 198 l; Gerstner and van Hemman, 1992), post-synaptic potential (Zetterberg et al., 1978; Lopes da Silva et al., 1974; Nunez, 1981), and a general level of activity within the system under study (Grossberg, 1973). Each of these comparisons are, in some sense, accurate in that the three measures of activity are related in a monotonic fashion, so that each may stand as a qualitative descriptor of the others. For quantitative analysis, however, a more precise understanding of network activity is vital. In the present work, close correspondence between the full model and the biological system resulted in a clear representation of biological aspects in the reduced model as well. In particular, understanding the relationship between firing rate, voltage, and synaptic drive allows for the proper incorporation of biological input data as well as the proper interpretation of the model's output. In Appendix D, we generalize this reduction process and present an alternate derivation of the Wilson-Cowan equations based on biophysical properties suggested by the particular reduction presented in Appendix B. Both firing rate and voltage have obvious biological interpretations. The question remains, however, as to whether synaptic drive, the key variable for which the

259

system is solved, can be interpreted in a biologically relevant way as well. As a measure of neural activity, synaptic drive takes into account the simultaneous role of each unit as both a pre- and postsynaptic element in the network. Formally, synaptic drive is the convolution of (presynaptic) firing rate with (postsynaptic) decay. Intuitively, the synaptic drive for a unit is obtained by considering the present activity (firing rate) within a population along with all previous activity, reduced appropriately based on when the event occurred (temporal decay). An interpretation for synaptic drive can be found by examining the units involved in the model's equation. For instance, the voltage contribution from RSU population activity to the FSU population is described by the term ll)ei eC S e, where Wei is the synaptic weight from the excitatory to inhibitory population measured in volts, and ec is the average number of excitatory synapses received by each population. In order for the full term to have units of volts, S e, the primary variable of the equation, must have units of 1~synapse, a dimensionless quantity per synapse, hence the term synaptic drive. Closely related to postsynaptic potential, synaptic drive is distinct in that it is independent of the strength of the connection between any two populations. By representing synaptic weights in terms of voltage, synaptic drive becomes a measure of the synaptic effectiveness of a population independent of subsequent voltage changes in the postsynaptic element. For instance, the drive of the FSU population measures the same whether acting on itself or acting on the RSUs. In this regard, synaptic drive is best envisioned as a measure of population activity that describes the influence of a given population on the overall behavior of the network. The overwhelming role of inhibition in network behavior becomes particularly apparent when measured in these terms because the level of excitability combined with a slow temporal decay acts to increase the synaptic drive of the FSU population to levels far above those of either RSUs or TCUs. The present model provides a minimal framework that incorporates four principle features of barrel organization: nonlinear neuronal properties, differential responsiveness of the RSUs and FSUs, convergence of thalamic input onto each population, and interconnections among and between RSUs and FSUs within the barrel. Recent cortical models of the visual system (Somerset al., 1995; Suarez et al., 1995) employ similar features with regard to excitatory and inhibitory

260

Pinto et al.

cells at various levels of complexity. The details of these and other cortical models inevitably depend on the particulars of the system under study, the questions addressed, and the nature of the stimuli presented. Nonetheless, we feel that these four properties effectively capture the essential operations of neocortical circuitry. An abstract formulation, such as presented here, can help to clarify those operations, and facilitate meaningful comparisons between distinct cortical systems.

Appendix A: Summary of Full Model

V~(t) = V{e(t - 1)e -1/~' - V{,i(t - 1)e -1/~ + ~ . weePe(V;(t))-]- £ j=l i

wtePt(VT(t))

.j=l

ic ~wiePi(V~(t))

k:

1...70

j=l

Vj(t) = Vj,e(t -- 1)e -1/~e - Vj, i(t - 1)e -1/rl ec

ICi

-]- y~weiPe(V;(t)) + E w t i P t ( V T ( t ) ) j=l

j=l

ic -- ~

llgiiei(vj(t))

k=1...30

j=l

V](t) : Membrane potential at time t for unit k of type x.

V~e(t): Contribution of excitatory synapses to V~(t) (similar for V~,i (t)) Px (V): Single unit activation function for cell type x, w i t h f o r m l / ( l + e

Tx: Temperature of activation function for units of type x: Te=2.5, p = - 6 0 mv: 0 = - 4 5 mv:

Resting membrane potential. Firing threshold.

Production of an action potential is based on probability generated by the activation function. When action potentials occur, a refractory period of 4 msec for excitatory cells and 3 msec for inhibitory cells is imposed.

Appendix B: Reducing the Full Model The reduction process begins with the full system in the form described in Appendix A, with two exceptions. First, the random generation of action potentials is excluded, maintaining instead the unit's firing rate as a continuous function of time. Second, the imposition of a refractory period on each unit is temporarily ignored but is later included as discussed in the text. The reduction procedure will be outlined only in the case of excitatory units as the inhibitory units proceed identically. Beginning with the published equations of the full system (Kyriazi and Simons, 1993) as presented in Appendix A the terms V~e(t - 1)e -l/~e and V~i(t - l) e -1/~' carry the residual effects of synaptic inputs over all previous times, and may be expressed as

V~ee(t -- 1)e-l/re = Z

tOee

s=0

rx ).

Ti=3.6

S

e -(t-s)/re

~

Wxy: Connection strength from x unit to y unit (mv): Wee = 1.0 4- 0.50, Wei = 1.0 + 0.50, Wie : 2.0 -4- 1.03, Wii :

+Z s=o

I~wtePt(Vf

(s))

e-(t-s)/r"

"

\j=l

1.5 -4- 0.75,

Wte = 3.9 4- 1.95, w , = 6.0 4- 3.00 xc: Number of synapses of type x received by

Similarly, V~,i(t - 1)e -1/~i becomes

V~,i(t- 1)e-l~ r,

each cell (convergence):

ec=42,

ic=12,

tCe =12,

tci = 1 0

E s~O

Wie

S

e-(t-s)/ri ) "

=

rx : Decay constant for type x PSP (msec): re=5,

Vi=15

The spatial and temporal series may be combined into a single term. Exchanging the order of summation and


replacing the temporal summation with an integral over time, the expression becomes

261

Wilson and Cowan (1972). Either method yields the relationship

dSf

V[(t) = Eec Wee fo' Pe(Vf(s))e-(t-s)/reds

+ s;(t) = Pe(V[(t)).

,j=l

+ j~l'=wte

Pt

The original 100 equations may now be expressed exclusively in terms of synaptic drive:

(S) e-(t-s)/reds

fo, (v?)

- i ic ~ Wie fot Pi(Vj(s))e-(t-s)/r'ds.

re--~7- (I) --~ S~(t)

= Pe Iz...~ ll)eeS.7(t) -~- Z WreST(I) \j= I

j=l

For each unit we define the synaptic drive, Sj (t), as The final step in the reduction entails actually reducing the number of equations. To do this, we introduce the averaged variables

S~(t) = f0 t P~(V[(s))e-(t-s)/reds = P e ( V f ( t ) ) * e -t/r"

i =-- Pi(Vj(t)) * e -t/r' Sj(t)

1

7o

se(t) =-- -~ ~

S T (t) -- P,(V? (t)) * e -'/r' ,

S~(t)

1 3O

so that the equations may be concisely presented in the form

s'(t)

=_

__

' S~(t)

k=l

ec

1

V~(t) = Z weeS;(t)

s

U

(t) -

s[ (t)

k=l

j=l

tCe ic "~ ~ wteSy (t) -- Z t°ieS}(t)" j=l

and then differentiate and substitute to obtain

j=l

dS e

Next, we move from a set of equations that represent the cell's voltage, V~(t), to equations which represent their synaptic drive, S~. As a first step, we move from voltage to firing rate, Pe(V~), simply by passing each side of the equation through the appropriate activation function:

Pe(V[(t)) = Pc

ll)eeS;(t) \j=l

Jr- j~__ltoteSy (t) -

j=l

)

wieS}(t) "

Taking advantage of the use of a simple exponential to describe the time course of PSPs, an expression for firing rate in terms of synaptic drive may be obtained from the definition either by differentiating with respect to time or applying the Laplace Transform and Convolution Theorem. This step is essentially equivalent to the time course graining technique presented by

Te--~ -~- L se(t ) re

= -~ k~1 Pe =

\j=l

WeeS~,j

Where k spans over the all units in a populatio n and j spans over the ec units connected to a given unit k. At this point an approximation must be used, introducing error into the model for the first time. It is necessary to approximate the average of a set of firing rates by the firing rate determined from the average voltage, that is,

7--0 k=l Pc(V[) ~ Pe ~-~ k=l For a sigmoid activation function, and only near the linear portion of the curve, this approximation is clearly valid only to first order. Methods to compensate for much of the remaining error are dealt with in the text.

262

Pinto et aL

With this substitution, the last term in the equation becomes to eeS~,j (t)

Pe k=l j = l 1

+ ~

70

t~c

1

ic

70

T __ "~ E ~ toieSk,j(t)~ WteSk,j(t) = j=l k-~l j=l /'

and each of the summation terms is dealt with as follows: 70

ec

Wee

70 k=l j = I

ec Sk'j(t) = ~

PSTH(t): Peri-stimulus time histogram obtained from pre-recorded barreloid neurons in response to whisker stimuli. Spike bins are scaled by number of contributing cells, the number of deflections per cell, and conversion factor ge. Conversion factor between firing rate and synaptic drive for population x: Ye ----5

xr:. Refractory parameter for population x:

Wee k=l ec

Yi = 15

er = 7 . 4

e'j(t)

ir = 4.1

j=l 70

Wee E S~(t) ec 70 k=l

Tx ' Temperature of activation function for population of type x: Te = 12.03

ec Wee se(t).

This successfully completes the reduction of the one hundred original equations to two, one each representing the average synaptic drive of the excitatory and inhibitory population:

T/ = 11.62

The remaining parameters take values identical to the full model as presented in Appendix A.

Appendix D: Generalizing the Reduction dS e

Te----~ - --[-se(t) = Pe (Wee ecSe (t) + WtetCe ST (t) -- WieicS i (t)) dS i

~'i d-"t- -t- S i ( t ) = Pi(weiecSe(t) + wtitciST(t) -- wiiicSi(t)).

Appendix C: Summary of Reduced Model dS e re d--T + se(t) = ye(1 -- erPe(Se)) Pe(weeecSe (t ) + WtetCeST (t ) -- Wiei cSi (t ) ) dS i ri d--7- + Si(t) = yi(1 - i r P i ( S i ) ) Pi (Wei ecS e (t) + wti tci S T (t) - toii i cS i (t)) dS r re---d-~- + S t ( t ) = PSTH(t). S x (t): Synaptic drive of population x at time t. Px (V): Activation function for population x with form ( I + E R F ( ~ ) ) / 2 , where

ERF(x), = - ~ f o e-Y2dY •

In this section we discuss, in more general terms, an alternate formulation of the equations describing activity in a local network population. Working exclusively with the excitatory population, it is clear that the voltage level of the population should be expressed in terms of post-synaptic potentials produced from the network. For a single action potential, the subsequent voltage change can be determined by an alpha function ~x (t), which describes the time course of individual PSPs (Rall, 1989). Given a continuous average firing rate, then, the voltage is expressed as the sum of excitatory and inhibitory PSPs, that is, v e ( t ) = Wee --Wie

(x)

Ote(t -- t ' ) P e ( V e ( ( ) ) dt'

Z

oo

o~i(t -- t ' ) P i ( V i ( t ' ) ) d t '.

Note that if we assume identical alpha functions for each population of the form or(t) = e ~ , then voltage may be expressed as,

ve(t) =

S

tt_t

e-i-(weePe(Ve(t'))-wiePi(Vi(t)))dt

oo

'.

A Quantitative Population M o d e l of W h i s k e r Barrels

and differentiated to obtain dV e "c tit = - - v e ( t )

q- w e e P e ( V e ( t ) ) -- t ° i e P i ( V i ( t ) ) "

A criticism o f the W i l s o n - C o w a n equations has been that the activation functions w e r e not separated in this fashion (Grossberg, 1973). In fact, both forms are correct but d e p e n d on whether the primary variable is m e a n t to represent voltage or, as seen below, synaptic drive. A n expression for firing rate is obtained simply by w r a p p i n g each side of the equation with the appropriate activation function: Pe(Ve(t))

-~ Pe

(S Wee

- Wie

~ e ( t -- t ' ) P e ( V e ( t ' ) ) d t

'

O0

oli(t - t ' ) P i ( g i ( t ' ) ) d (

A n d synaptic drive, defined as the convolution o f firing rate with synaptic decay, is constructed accordingly: se(t) =

// //

C~e(t -- t ' ) P e ( V e ( t ' ) ) d t '

O0

=

f f e ( t -- t ' ) P e ( w e e S e ( t ') - w i e S i ( t ' ) ) d t '

OG

N o t e the p l a c e m e n t of the activation function as c o m pared to the voltage equation presented above. Finally, if we again assume an alpha function with instantaneous rise and exponential decay (i.e., ofe = -¢ e ~ ) , w e m a y differentiate to obtain equations identical to those o f W i l s o n and Cowan, excluding refraction: dS ~ 75e

-~_ - s e (t) -[- Pe(toeeSe (t) - toieSi (t) ).

dt

Acknowledgments Special thanks ful discussions data files used N S 19950, N S F

to Harold T. Kyriazi for m a n y fruitand for p r o v i d i n g access to thalamic in the full model. Supported by N I H I B N 9 4 2 1 3 8 0 , and N S F - 9 3 0 3 7 0 6 .

References Anninos E Argyrakis P (1983) A mathematical model for the decay of short-term memory with age. J, Theol: Biol. 102:191-197.

263

B arnwell GM, Stafford FS (1977) A mathematical model of decisionmaking neural circuits controlling behavioral mode selection. Bull Math. BioL 39:223-237. Bertholon IF, Caries J, Eugene M, Labeyrie E (1988) A dynamic analysis of the ventilatory response to carbon dioxide inhalation in man. 3i PhysioL 398:423-440. Brannan JR, Boyce WE (1981) Spatially localized interactive neuronal populations. I. A mathematical model. Bull. Math. BioL 43:427-446. Britton NF, Skevington SM (1989) A mathematical model of the gate control theory of pain. J. Theor. Biol. 137:91-105. Brumberg JC, Pinto DJ, Simons DJ (1995) Inhibitory gradients and spatial summation in barrel cortex (Abstract). Neurosci. Abstt 21 :120. Carvell GE, Simons DJ (1995) Task- and subject-related differences in sensorimotor behavior during active touch. Somatosensory Mot. Res. 12:1-9. Cope DK, Tuckwell HC (1979) Firing rates of neurons with random excitation and inhibition. J. Theor Biol. 80:1-14. Douglas RJ, Koch C, Mahowald M, Martin KAC, Suarez HH (1995) Recurrent excitation in neocortical circuits. Science 269:981985. Ermentrout GB (1995) Phase-plane analysis of neural activity. In: MA Arbib et al., eds. Handbook of Brain Theory and Neural Networks. MIT Press, Cambridge, MA. pp. 732-735. Ermentrout GB (1994) XPP:X-windows phaseplane--dynamical systems software, ftp://mthbard.math.pitt.edu/pub/bardware/xpptut/start.html. Ermentrout GB, Kopell N (1991) Multiple pulse interactions and averaging in coupled neural oscillators. J. Math. Biol. 29:195217. Ermentrout GB, Troy WC (1989) The uniqueness and stability of the rest state for strongly coupled oscillators. SIAMJ. Math. Anal. 20:1436-1446. Evans ER (1989) Random dilution in a neural network for biased patterns. J. Phys. A 22:2103-2118. Gerstner W, van Hemmen JL (1992) Universality in neural networks: The importance of the "mean firing rate". Biol. Cybern. 67:195205. Ginzberg I, Sompolinsky H (1994) Theory of correlations in stochastic neural networks. Phys. Rev. E 50:3171-3192. Grossberg S (1973) Contour enhancement, short term memory, and constancies in reverberating neural networks. Stud Appl. Math. 52:213-257. Hasselmo ME, Bower JM (1992) Cholinergic suppression specific to intrinsic not afferent fiber synapses in rat piriform (olfactory) cortex. J. NeurophysioL 67:1222-1229. Hasselmo ME, Schnell E (1995) Laminar selectivity of the cholinergic suppression of synaptic transmission in rat hippocampal region CAI: Computational modeling and brain slice physiology. J. Neurosci. 14:3898-3914. Hoffman GW (1986) A neural network model based on the analogy with the immune system. J. Theor. Biol. 122:33-67. Ingber L (1982) Statistical mechanics of neocortical interactions: I. Basic formulation. Physica D 5D:83-107. Ito M (1985) Processing of vibrassa sensory information within the rat neocortex. J. NeurophysioL 54:479--490. Kawahara T (1980) Coupled Van der Pol oscillators: A model of excitatory and inhibitory neural interactions. Biol. Cybern. 39: 37-43.

264

P i n t o et al.

Keller A (1995) Synaptic organization of the barrel cortex. In: EG Jones, IT Diamond, eds. Cerebral Cortex. Pleneum Press, New York, Vol. 11. pp. 221-262. Klaasen GA, Troy WC (1984) Stationary wave solutions of a system of reaction-diffusion equations derived from the FitzhughNagnmo equations. SlAM J. Appl. Math. 44:96-I 10. Kleinfeld D, Raccuia-Behling F (1990) Circuits constructed from identified Aplysia neurons exhibit multiple patterns of persistent activity. Biophys. J. 57:697-715. Kyriazi HT, Carvell GE, Brnmberg JC, Simons DJ (1996) Quantitative effects of GABA and bicuculline methiodide on receptive field properties of neurons in real and simulated whisker barrels. J. NeurophysioL 75:547-560. Kyriazi HT, CarveU GE, Simons DJ (1994) OFF response transformations in the whisker/barrel system. J. Neurophysiol. 72:392401. Kyriazi HT, Simons DJ (1993) Thalamocortical response transformations in simulated whisker barrels. J. Neurosci. 13:16011615. Lagerlund TD, Sharbrough FW (1989) Computer simulation of the generation of the electroencephalogram. EEG & Clin Neurophys. 72:31-40. Land PW, Buffer SA, Jr., Yaskosky JD (1995) Barreloids in adult rat thalamus: Three-dimensional architecture and relationship to somatosensory cortical barrels. J. Comp. Neurol. 355:573-588. Lockery SR (1993) A neural-network model of sensorimotor transformations in the local binding reflex of the medicinal leech. Am. Zoo. 33:40-53. Lopes da Silva FH, Hocks A, Smits H, Zetterberg LH (1974) Model of brain rhythmic activity: The alpha-rhythm of the thalamus. Kybernetik. 15:27-37. McCormick DA, Connors BW, Lighthall JW, Prince DA (1985) Comparative electrophysiology of pyramidal and sparsely spiny stellate neurons of the neocortex. J. Neurophysiol. 54:782-806. Moser E, Mathiesen I, Andersen P (1993) Association between brain temperature and dentate field potentials in exploring and swimming rats. Science. 259:1324-1326. Matsushita T, Moriyama S, Fukai T (1995) Switching dynamics and the transient memory storage in a model enzyme network involving Ca2+/caimodulin-dependent protein kinase II. BioL Cybern. 72:497-509. Nicolelis MAL, Lin RCS, Woodward D J, ChapJn JK (1993 ) Dynamic and distributed properties of many-neuron ensemble in the ventral posterior medial thalamus of awake rats. Proc. Natl. Acad Sci. USA. 90:2212-2216. Nunez PL (1981) A study of origins of the time dependencies of scalp EEG: ii-Experimentai support of theory. IEEE Biomed. 28:281288. Paydarfar D, Eldridge FL, Kiley JP (1986) Resetting of mammalian respiratory rhythm: Existence of a phase singularity. Am. Z Physiol. 250:R721-727. Poston T, Stewart I (1978) Nonlinear modeling of multistable perception. Beh. Science. 23:318-334. Press WH, Flannery BP, Teukolsky SA, Vetterling WT (1988) Numerical recipes in C: The art of scientific computing. Cambridge University Press, New York.

Rail W (1967) Distinguishing theoretical synaptic potentials computed for different soma-dendritic distributions of synaptic input. J. Neurophysiol. 30:1138-1168. Rail, W (1989) Cable theory for dendritic neurons. In: C Koch, I Segev, eds. Methods in neuronal modeling. MIT Press. Cambridge, MA. pp. 9-62. Recanzone GH, Merzenich MM, Shreiner CE (1992) Changes in the distributed temporal response properties of SI cortical neurons reflect improvements in performance on a temporally based tactile discrimination task. J. NeurophysioL 67:1071-1091. Sejnowski TJ (1976) On global properties of neuronal interaction. Biol. Cybern. 22:85-95. Simons DJ (1978) Response properties of vibrissa units in rat SI somatosensory neocortex. J. Neurophysiol. 41:798-820. Simons DJ (1985) Temporal and spatial integration in the rat SI vibrissa cortex. ,L NeurophysioL 54:615-635. Simons DJ (1995) Neuronal integration in the somatosensory whisker/barrel cortex. In: EG Jones, IT Diamond, eds. Cerebral Cortex. Pleneum Press, New York. Vol, 11. pp. 263-297. Simons DJ, Carvell GE (1989) Thalamocortical response transformation in the rat vibrissa/barrel system. J. Neurophysiol. 61: 311-330. Skondras M (1988) RF-cell: A model for populations of randomly interconnected neurons. Bull. Math. Biol. 50:43-66. Somers DC, Nelson SB, Sur M (1995) An emergent model of orientation selectivity in cat visual cortical simple cells. J. Neurosci. 15:5448-5465~ Suarez H, Koch C, Douglas R (1995) Modeling direction selectivity of simple cells in striate visual cortex within the framework of the canonical microcircuit. Z Neurosci. 15:6700-6719. Sugitani M, Yano J, Ooyama H (1990) Somatotopic organization and columnar structure of vibrissae representation in the rat ventrobasal complex. Exp. Brain Res. 81:346-352. Tolbert D1, Bantli H, Bloedel JR (1976) Anatomical and physiological evidence for a cerebellar nucleo-cortical projection in the cat. Neurosci. 1:205-217. Welker C (1976) Receptive fields of barrels in the somatosensory neocortex of the rat. J. Comp. Neurol. 166:173-189. White EL (1989) Cortical circuits: Synaptic organization of the cerebral cortex. Structure, Function, and Theory. Birkhanser, Boston MA. Wilson HR, Cowan JD (1972) Excxtatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 12:124. Wilson M, Bower JM (1992) Cortical oscillations and temporal interactions in a computer simulation of piriform cortex. J Neurophysiol. 67:981-995. Woolsey TA, Van der Loos H (1970) The structural organization of layer IV in the somatosensory region (SI) of mouse cerebral cortex. Brain Res. 17:205-242. Wright JJ, Kydd RR, Lees GJ (1987) Gross electrocortical activity as a linear wave phenomenon with variable temporal damping, regulated by ascending catecholamine. Int. J. Neurosci. 33: 1-13. Zetterberg LH, Kristiansson L, Mossberg K (1978) Performance of a model for a local neuron population. Biol. Cybern. 31:15-26.