Modelling Executive & Attentional function in Alzheimer's Disease - 1999 MSc Thesis - Dr. M. Moutoussis

IV. Methods

In order to test whether candidate models accounted for the relevant experimental data the following methodology was pursued. First, the literature was kept under regular survey. Second, important principles of model structure were operationalised. Third, suitable programming platforms were chosen and a simulation strategy exploiting the philosophy of these platforms was outlined. Fourth, a hierarchical programming strategy allowed implementation of increasingly complex models. Fifth, auxiliary programming tools were developed for the handling of input and output of models. Sixth, a quality control strategy was implemented to minimise the risk of programming errors.

 

1. Methods used for Systematic Review of literature     Not available online - Please e-mail me for details

2. Criteria for setting model structure         Not available online - Please e-mail me for details

3. Effective use of programming environments         Not available online - Please e-mail me for details
 

4. Core model structure and programming
 

A system of interconnectable modules (that can represent neurons or other connectionist 'units' such as lumped cortical areas) with their connections ('dendritic trees'), outputs ('axons') and state variables, was developed (Appendix I). These 'units' could be driven by arbitrary dynamics to specify how they respond to their inputs so as to produce outputs. Classes were programmed to handle time-dependent equations to drive the 'units'. Classes used for solving systems of ordinary differential equations (ODEs) were based on an adaptive step, fourth order Runge-Kutta method (Press et al, 1988). Classes for return maps were also programmed.
 
 

Modelling emulated the progression from single task to dual task experiments and examined the effects of attention and concurrent task on performance. The evolving structure permitted exploration of the simplest models to help determine and the most plausible characteristics of subsequent ones.
 
 

As the Baddeley experiments that I aimed to simulate do not report detailed discrimination of features within each sensory system (e.g. detection of a tone, rather than discrimination between tones, is used), each cortical area was simulated by a 'lumped' unit. Apart from this being common practice, both psychological (Houghton's work) and physiological (Freeman, 1992) demonstrations have been provided of the validity of treating a cortical area as a 'lumped' neural mass for the purposed of simulation of aspects of its overall behaviour, such as its overall activation. The cortical area corresponding to each of the two sensory modalities in a dual task was initially simulated by an 'attentional triad' consisting of one 'perceptual' and two 'gain control' units (fig. 4; cf. fig. 2a field C).
 
 

Figure 4: Models directly derived from the attentional control theory of Houghton & coworkers . Models use 'attentional triads' (gain units labelled + and -) as lumped representations of perceptual cortices. a. Direct lateral inhibition; also synapse and cortical area labels, which also apply to (b.) and (c.) b. Descending control of attention only. c. Descending and ascending connections forming a 'match' mechanism.

In successive experiments, the two cortical areas were connected by first, Direct Lateral Inhibition (DLI) only; then, descending attentional control (DAC); and thirdly, a 'Match' attentional system (MAS). This last stage represents the model testing the core hypothesis of this study, that an attentional control system featuring matching between target and percept, plus cross-modality inhibition, can simulate the patterns of performance during dual task experiments in normal and, when modified, in AD subjects. Based on this model alternative units for the attentional triad and matching mechanisms, diverging from the Houghton models, were explored. The three basic simulation steps are now described in more detail.

 

1. Direct Lateral Inhibition only. It is thought (Nunez, 1995) that most inter-area corticocortical connections are excitatory. Within the Houghton model a lateral inhibitory effect is however easily implemented through excitation by one area of the inhibitory gain units of the other (fig. 4a).
 

2. Descending attentional control. Descending influence from a task-specific attentional area provides excitatory input to its 'own' and inhibitory input to the 'opposite' modality (fig. 4b), according to the principle of cross-modality inhibitory attentional control.
 

3. Descending and ascending projections implement 'match' loops. As there were no discrimination tasks involved, the necessity to consider 'mismatch' loops did not arise. However, the issue of the details of implementation the 'match' units did arise. Rather than the units being programmed to directly perform logical functions, as in the original Houghton models, the functionally approximate but physiologically more plausible use of a simple logistic sigmoid neurons, such as in fig. 2, was adopted. Parameters for these are not available either from physiology or previous work. They were therefore initially set to be such that for the simulation of normals attending to a single task, the presentation of the stimulus changed 'match' unit activation from a low (~10% of max), resting state to a high (~90% max), activated state. For AD, the matching mechanism was impaired; there was no a priori reason to suggest that such impairment should affect the excitatory output of the mechanism more than its inhibitory output. AD was therefore assumed not to affect their balance but only diminish their output.
 

Alternative explorations first examined the effect of DLI coexisting with MAS (fig. 5) and simulating attentional triads consisting of units more representative of the physiological properties of cortex.
 
 

Figure 5: Combination of Direct Lateral Inhibition and a 'match' mechanism.
Finally an alternative form of the matching process, the Oscillatory Match Model (OMM; see Discussion) was explored. Here two neural masses capable of oscillation were interlinked. Their coupling could lead to synchronisation and possibly amplification of activity, and these are taken to indicate 'matching'. The process was inspired from the extensive work of Kelso (1995) on coupled oscillator systems (fig. 6).

Figure 6a: Oscillatory model of a 'match' mechanism; Adaptation of the dual pathway model of fig 3. to include oscillatory activity. Cross-modality interference is now not associated with descending control but with local effects, as per interpretation of the lateral inhibition findings. This is shown by the asymmetric influence of the 'match' modules of each pathway on each other (grey arrows)
 
 
 

The OMM was investigated in simulated models with the following structure.  [ Only a schematic description given here - Please e-mail me for details  ] Two KI sets (Freeman, 1975) interlinked by negative feedback were used to simulate each oscillator (each of the two circles at the top of the diagram of fig. 6). The equation governing the state variable x(i) of the i-th KI set is:

 

d2x(i)/dt2 + A*dx(i)/dt + B*x(i)

= S(j) {I(j)} + S(n) { Kni*Q(n)[x(n)] }, Relation (Rln) 1
 
 

For simulation of the interfering modality a source of  broad-band, brain-activity-like 'noise' was required.  The interfering noise was therefore simulated a system displaying '1/f' spectrum oscillations in the form of Sil'nikov chaos (Kelso & Fuchs, 1995). The model is shown in fig. 6.

Figure 6b. OMM reduced to its essentials for preliminary study.
 
 


5. Auxiliary programming & software          Not available online - Please e-mail me for details
6. Program Quality control          Not available online - Please e-mail me for details

 

7. Mathematical Analysis
 

Problematic simulation results were investigated both analytically and graphically, using dynamical systems methods (Abraham & Shaw, 1992, Devaney, 1989). The utilisation of reduced systems of return maps to investigate time-dependent behaviour of biological systems was inspired by the work of Chialvo & coworkers (Chialvo et al, 1990), while the use of simple linear methods for stability analysis of ODE systems followed Glass & Mackey (1988).
 

To understand the performance of Houghton's attentional triad, use was made of :

a. Algebraic determination of steady-state model solutions via setting the rate of change of all variables representing neuronal activities equal to zero.

b. Determination of stability of solutions thus obtained by deriving linear approximations to the system equations and considering their evolution near the steady-states.

c. A simplified map preserving key model properties was derived, thus obtaining a return-map, rather than differential-equation, model.

d. Graphical analysis of the simplified map was performed to determine the characteristics of its steady states.
 


Modelling Executive & Attentional function in Alzheimer's Disease - 1999 MSc Thesis - M. Moutoussis

V. Results

1. Models directly based on the Houghton attentional control theory

 

A. Direct lateral inhibition (DLI) model
 

This was used to examine the response of a lumped cortical area under the influence of an external signal, usually inhibitory, when an appropriate stimulus is presented. This mimics the response of Auditory areas to a brief tone, on a background of the constant interference due to Visual (tracking) processing. It was found that the baseline state of the secondary task unit was typically reduced by 50% of its maximum response under single task conditions. The maximum response itself was reduced by 20% only, while the difference in the timing of the response was very small and depended on the risetime of the stimulus (figures given here refer to the following set of parameters: Ext. stimulus I=0.75; W1=0.5; W+ = 0.3; W- = -0.3; D=0.4; Wi(Vis->Aud)=0.7; Wi(Aud->Vis)=0.35. The exact percentages depend on the parameters used, but their comparative relations don't: for example, the effect on the baseline is always stronger than that on maximum activation).
 
 

As the onset of response to the Auditory signal was little affected by the presence of an inhibiting, concurrent task, it was decided, following Cohen et al (1992), to take the cumulative activation of a pathway rather than its risetime behaviour to correspond to behavioural reaction time.
 
 

Model behaviour on stimulus offset was anomalous, units sometimes remaining 'switched on' after stimulus offset. This difficulty was also encountered in the much more complicated Houghton models of attention, as described in the introduction. Here further use of the model was limited to parameter ranges where stimulus offset is accompanied by decay of cortical activation. The matter is investigated analytically as follows.
 
 

Following Houghton, we consider the limit where the contribution of inhibitory gain units is negligible, for example because some external cause has switched that unit off (the attending dyad approximation). The equations of the system are :
 

                             +
da /dt = -D*a + ( 1 - a ) * W * a           } Rln. 2a
  p          p         p         on         }
 
da  /dt = -D*a  + ( 1 - a  ) * W1 * [ a ]   } Rln. 2b
  on          on         on            p
 

 

where a is a variable describing the state of each unit, D the time decay constant of the units and Wx is the weight of synapse x (as in fig. 4a). The index 'on' refers to the excitatory gain unit, while 'p' to the perceptual unit. [x] = 0 if x <= 0, while [x] = x if x > 0. For the equilibrium state, setting both Relns. 1 = 0 gives :
 

         +       2
       W1 * W - D
a  = ----------------
 p                +
     W1 * ( D + W   )      } Rln. 3
Stability analysis shows this solution to be stable if it is positive, which is guaranteed if

 
      +    2
W1 * W  - D    > 0, } Rln. 4
 

 

i.e. the synaptic weights around the positive feedback loop dominate over the decay constant. This explains why the units of the Houghton model, arranged in 'attentional triads', can get stuck in an activated state. It can be further proven that an attentional triad with inputs to each unit has single unique equilibrium solution if inhibitory drive is adequate to drive the activation of the perception unit negative and is potentially multistable for positive driving.
 
 

A simplified return map model was constructed for the attending dyad (or triad) by observing that for da/dt = 0, each unit in the triad will have an activation governed by its steady-state input-output curve,
 

a = I / (I + D) , } Rln. 5
 
 

where I is any positive input. This curve is convex upwards. The construction of the map (fig. 7) confirms the analytical results of the ODE model and suggests that they can be overcome by the adoption of activation curves which at the point of zero input are concave upwards (fig. 8).
 
 

Figure 7 a. We examine an 'attending' state where some signal has completely suppressed the Off-gain unit. Results would be qualitatively similar whenever there the On unit is more active than the Off. b. An input I arrives at time t=0 and excites the Property unit to a level p(0)=P(I). [The activation of the Property unit at time t is p(t) ; the input/output (i/o) function of the same unit for input i is P(i); similar notation for the On unit.] In the next time step t=1activity propagates to the On unit and produces an activation on(1)=On(p(0)) . c. The On unit i/o function On(i') is plotted 'on its side', with the origin of axis at the point (p,i)=(0,I) . This allows visualisation of how p(0) feeds into the On unit and produces on(1) , and in turn how on(1) adds to I and feeds into the Property unit, causing p(2)=P(I+on(1)). The process then repeats itself, p(2) producing on(3) etc. (This analysis can easily be extended to incorporate the Off unit). The activation of the two units is thus represented by a 'staircase' between the two curves P(i) and On(i') producing a so-called 'return map', a function giving the state of the system s(t) in terms of its state s(t-1). The system will come to rest at a point where the two curves cross. If we were to put the system there to start with, it would not move: these crossings are equilibrium points. If the 'staircase' leads towards such a point, the latter is an attractor (a point attractor); if away from it, a repellor. It can now be examined whether model properties correspond to real life. The resting state (the origin of axes) of this 'attending' system is a repellor: the perceptual units can be activated in the absence of stimulus !

 
 

Figure 8: As in fig. 7, but a shallow gradient near the origin guarantees low output for low input. This in turn allows activation of the On unit to result in a true increase in gain of Property unit output.

 

A given increase in the drive of an off-gain unit always has a larger impact on reducing perceptual unit activation than an equal increase in the drive of an on-gain unit, as indeed one might expect by inspection of the general state equation for a unit:
 

                            +          -
da/dt = -D*a + ( 1 - a ) * I - (1+a)* I } Rln. 6
Where I+ and I- are the excitatory and inhibitory inputs respectively (both positive). Further, in a simplified triad with all synaptic weights equal to w, and input to the perceptual unit only, the ratio w/D limits the maximum value that the perceptual activation can attain. Increasing w/D reduces the equilibrium value attained by perceptual unit and makes its input - response curve flatter.

 

In summary, the simplest models reveal parameter ranges that have to be avoided in more complex simulations. The DLI model is however sufficient to demonstrate how a very simple mechanism could explain cross-modality interference.
 
 

B. Descending attentional control (DAC)
 

This is shown in fig. 4b. Attention is mediated by modality specific top-down activation. This increases response in its own, 'attended', modality and reduces cross-modality activation. Thus this is the minimal model of the effect of attention in suppressing non-attended modalities. Top-down activation or inhibition again affects baseline activation more than response plateau. In an experiment, for example, where the visual task is ongoing, but the subject concentrates on detecting the auditory tone when this is to arrive, the baseline visual state is suppressed by about 80% of the maximum it then attained when the visual task was under way (from 0 to -.35 in arbitrary activation units; visual task response was +0.42 ). The auditory baseline activation increases instead by 38% of its task activation level (from 0 to +0.22; auditory task response +0.58). The precise figures depend on the parameters used, but if chosen not to fulfil Rln. 4 - i.e. not to have any spurious stable states - the pattern of results remains robust. Note that this includes a greater activation, in absolute terms, for the attended modality compared to the non-attended one, but a lesser activation (lesser gain) if the baseline state is taken into account (fig. 9).
 

Thus the DAC model simulates the normal pattern of cross-modality interference, and has some success in simulating the qualitative pattern of increased performance for an attended modality. However, as there are no ascending connections the cross-modality effects are independent of stimulus processing and thus the model does not simulate the Baddeley experiments.
 
 

Figure 9: Simulation of descending attentional control. Panels a & b show perceptual unit activation, c & d the input to the respective modalities. Vertical axes are calibrated so that peak and trough values can be read off the graphs, rather than compared visually. This format is retained in all figures. A short auditory tone is presented during a prolonged visual task. Note the effect of attention on baseline activation.

 

C. Descending and ascending 'match' attentional system (MAS)
 

This model (fig 4c) affects lateral inhibition indirectly, and such inhibition is shown to be dependent both on descending (attentional) activation of the cross-modality perceptual unit and on the presence of the cross-modality stimulus. This is the minimal model that simulates features of the dual-task performance in normal people.
 
 

The basic function of the model is shown in fig. 10. In the first part of this figure (a) both stimuli of the dual task are presented, but only one is attended. This is achieved by setting the 'target activation' of the auditory but not the visual pathways to a high value. It is evident that attention changes baseline activation even in the absence of stimuli (cf. behaviour near time = 50). Successful properties of the system include first, peak activation of the attended system being higher than of the non-attended one (0.51 vs. 0.48 in this example: a small difference); and second, a cross-modality effect evident by a drop in visual activation by about 15% of its peak value during the time when both stimuli are present. Fig. 11b shows model behaviour under true dual task conditions, i.e. when attentional activation is present for both modalities. As it should, the model shows greater activation of the visual modality than in fig 10a, and lesser activation of the auditory modality. If we take cumulative activation to correspond to performance, the reduction in area-under-the-curve between the two figures (shaded) is 17%, consistent with experiment in order of magnitude terms. No attempt was made here to simulate the fact that the primary task affects the secondary more than vice versa in most of Baddeley's experiments - this would be straightforward.

Figure 10: Dual-task model relying on 'matching', for normal subjects. In this case however only the auditory modality is attended (only the auditory match unit receives top-down activation). The shaded area is taken to be a measure of performance in the task. However due to the shape of the response this is approximately proportional to the plateau value.
Figure 11: Dual-task model with same structural parameters as in fig. 10, but now both modalities are attended (both match units receive top-down activation).
The relative effect of dual task performance in AD can be successfully simulated by this system, through making the match units less efficient. This is simulated here by reducing the top-down activations, (T1 and T2 in fig. 4c). A relatively successful simulation example is shown in figs 12 and 13. Here Normals have better performance than AD in any task, Dual task has worse performance than Single for any group, and the relative drop in performance during dual task is greater in the AD than in the Normals. Further reduction in the drive to the 'match' units, T1 and T2, decreases dual task performance more. Some differences are however very small, unlike Baddeley's experiments; and reduction in T1/T2 soon ceases to further reduce dual task performance, the relative performance bottoming out at about 0.74. Furthermore, this general qualitative congruity with experiment is only present for fairly specific parameter ranges and is not part of the structure of the model as such. Even then, quantitative match is not achieved, AD -Normal activation differences being in general too small to account for experimental results. Indeed, detailed parameter exploration shows that it is more common for anomalous results to appear (e.g. AD increasing perceptual unit activation or reducing dual task related drop in performance). If the MAS is to be used as basis for further dual task simulations development must be limited to parameter ranges that do not exhibit such anomalies. These include low stimulus amplitudes and a high ratio of synaptic weights of the own-modality over the cross-modality projections.

 

2. Direct lateral inhibition plus desceding/ascending match model
 

The simplest alteration of the basic MAS model that will provide a robust effect of increasing the dual task effect when the efficiency of the matching mechanism is impaired is to combine the DLI and MAS systems. Simulation of this system allows quantitative match with the Baddeley experiments over a large range of parameter values.

Figure 12: Simulation of the performance of normal subjects under dual task conditions. The parameters in this and the next figure have been selected to optimise accordance with experimental results. Note, for example, the lesser amplitude of the input stimuli.
Figure 13: Simulation of performance of AD subjects. This is achieved by decrease in the performance of the match mechanism, here due to reduced top-down stimulation. Other parameters are identical to fig. 12. Detailed comparison of output levels with fig. 12 shows qualitative agreement with the Baddeley results.

 

3. Revised model - preliminary results
 

A. Alternative neural unit equations
 

Simulations were performed while substituting the Grossberg equations on which the original Houghton model is based with the Freeman 'KI' equations. The DAC model was implemented using the KI equations, and it was confirmed by simulation that activation of the on-gain unit by positive attentional input here not only increases activation on presentation of a stimulus, but also the difference between baseline and response activations, thus truly increasing the gain of the triad. A second result is although high synaptic weight products around the positive feedback loop can still result in spuriously activated states, the system is not as sensitive to this effect, again as predicted by graphical analysis (fig 8). Secondly, activation of attentional triads made of KI units can easily be made to oscillate, consistent with physiological studies. Finally, KI units respond much more slowly to step inputs than the original Grossberg units. They may thus lend themselves more readily to investigation of time-dependent phenomena such as reaction times.
 

As an example, the set of W1 = W+ = - W- = 0.7, T1 = 0.2 T2 = 0 produces a baseline activation of 0.17, increasing to 0.83 on stimulus presentation. When T1 = T2 = 0, baseline activation is 0 and response to stimulus 0.5. In this example the increase in gain is about 30%.
 
 

B. Alternative matching mechanism
 

Here the criterion for a successful 'match' is the synchronised activation of `perceptual / object field' and `association / match' areas, rather than just the level of activation of these areas. In the normal case the models shows synchronisation dependent on first, the presence of a stimulus and second, the presence of descending ('target') activation (data not shown). In the case of dual task synchronisation is again achieved, but not as smoothly (fig. 14); for high levels of interference the speed with which it is achieved varies with the precise characteristics of the interfering noise within each trial. Also achieved is the establishment of a cooperative oscillation with increase in the overall level of activity of the cooperating areas much as in the models above.

In the case of AD and in the absence of interfering activity (single task condition), synchronisation appears to be achieved more slowly and the cooperative activity has lesser magnitude, consistent with a decreased performance (fig. 15). This was suggested by preliminary exploration of the model. The introduction of interference (dual task) in the case of AD can disrupt synchronisation gravely or even abolish it completely (consistent with 'missed responses' in Baddeley's experiments) . This model therefore appears to have the potential to simulate the progression of AD from a prolongation of reaction times to complete task failure.  Full exploration of this promising model has been carried out. Please contact me for details

Figure 14: Simulation of dual task interference for normal controls (Revised model) a., b. Response of the coupled cortical areas of fig. 6b,c to a stimulus presented at time t=20 and withdrawn at t=60. The 'match' units are subject to top-down, 'attentional' activation. Note that during this period an oscillation of gradually increasing magnitude is produced cooperatively between the two areas. c. Noise represents cross-modality coupling (therefore interference) d. Estimate of inter-areal synchronisation. This is shown for clarity for the period of stimulus presentation only. The timing (in units of phase angle, rad) of each peak-to-peak wave of the two areas is compared. The zero, flat baseline describes absolute synchronisation and is drawn for purposes of demonstration only - it is not calculated from oscillations outside the stimulus epoch ! The graph shows that at the start of stimulus the association / match area lags behind by about 1.9 rad (a random value) but quickly converges to a near-zero (synchronised) phase value. The irregular path to convergence is because of the noise.

Figure 15: Simulation of AD during single task performance, showing a. the response of the association area and b. synchronisation (cf. fig 14b & d). AD is simulated by a reduction in the inter-area coupling Kxx (cf. fig 6c). A cooperative oscillation is again produced, but is of smaller amplitude. There is no interfering modality, so the system smoothly but slowly converges to a near-zero phase value.


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