Lecture Notes in Computer Science
A Computational Model of Formation of Grid Field and
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- Keywords
- 2.1 A Module of Local Path Integrator
- 2.3 Theta Phase Precession in the Grid Field
- 3 Mathematical Formulation
- 4 Computer Simulation of Theta Phase Precession
- 5 Discussions and Conclusion
A Computational Model of Formation of Grid Field and Theta Phase Precession in the Entorhinal Cells Yoko Yamaguchi 1 , Colin Molter 1 , Wu Zhihua 1,2 , Harshavardhan A. Agashe 1 ,
and Hiroaki Wagatsuma 1
1 Lab For Dynamic of Emergent Intelligence, RIKEN Brain Science Institute, Wako, Saitama, Japan 2 Institute of Biophysics, Chinese Academy of Sciences, Beijing, China yokoy@brain.riken.jp Abstract. This paper proposes a computational model of spatio-temporal property formation in the entorhinal neurons recently known as “grid cells”. The model consists of module structures for local path integration, multiple sensory integration and for theta phase coding of grid fields. Theta phase precession naturally encodes the spatial information in theta phase. The proposed module structures have good agreement with head direction cells and grid cells in the entorhinal cortex. The functional role of theta phase coding in the entorhinal cortex for cognitive map formation in the hippocampus is discussed.
cell.
1 Introduction In rodents, it is well known that a hippocampal neuron increases its firing rate in some specific position in an environment [1]. These neurons are called place cells and considered to provide neural representation of a cognitive map. Recently it was found that the entorhinal neurons, giving major inputs to the hippocampus fire at positions distributing in a form of a triangular-grid-like patterns in the environment [2]. They are called “grid cells” and their spatial firing preference is termed “grid fields”. Interestingly, temporal coding of space information, “theta phase precession” initially found in hippocampal place cells were also observed in grid cells in the superficial layer of the entorhinal cortex [3], as shown in Figs.1 – 3. A sequence of neural firing is locked to theta rhythm (4~12 HZ) of local field potential (LFP) during spatial exploration. As a step to understand cognitive map formation in the rat hippocampus, the mechanism to form the grid field and also the mechanism of phase precession formation in grid cells must be clarified. Here we propose a model of neural computation to create grid cells based on known property of entorhinal neurons including “head direction cells” which fires 152 Y. Yamaguchi et al. when the animal’s head has some specific direction in the environment. We demonstrate that theta phase precession in the entorhinal cortex naturally emerge as a consequence of grid cell formation mechanism.
Fig. 1. Theta phase precession observed in rat hippocampal place cells. When the rat traverses in a place field, spike timing of the place cell gradually advances relative to local field potential (LFP) theta rhythm. In a running sequence through place filed A-B-C, the spike sequence in order of A-B-C emerge in each theta cycle. The spike sequence repeatedly encoded in theta phase is considered to lead robust on-line memory formation of the running experience through asymmetric synaptic plasticity in the hippocampus.
(EC deeper layer and EC superficial payer) and cortical areas giving multimodal input. Theta phase precession was initially found in the hippocampus, and also found in EC superficial layer. EC superficial layer can be considered as an origin of theta phase precession.
A Computational Model of Formation of Grid Field and Theta Phase Precession 153 HC place cell EC grid cell EC LFP theta Time
hippocampal place cell. Bottom) Theta phase precession observed in the EC grid cell and in the hippocampus place cell.
Firing rate of the ith grid cell at a location (x, y) in a given environment increases in the condition given by the relation:
= α i +
i cos
φ i +
i cos(
φ i + π / 3),
= β
+ nA i sin φ
+
φ
+ π
with
= integer
+ r ,
(1)
where
φ i , A i and (
α i , β i ) denote one of angles characterizing the grid orientation, a distance of nearby vertices, and a spatial phase of the grid field in an environment. The parameter r is less than 1.0 giving the relative size of a field with high firing rate.
154 Y. Yamaguchi et al.
entorhinal cortex. The bottom layer consists of local path integration module with a hexagonal direction system. The middle layer associates output of local path integration and visual cue in a given environment. The top layer consists of triplet of grid cells whose grid fields have a common orientation, a common spatial scale and complementary spatial phases. Phase precession is generated at the grid cell at each grid field.
The computational goal to create a grid field is to find the region with n, m = integer + r . We hypothesize that the deeper layer of the entorhinal cortex works as local path integration systems by using head direction and running velocity. The local path integration results in a variable with slow gradual change forming a grid field. This change can cause the gradual phase shift of theta phase precession in accordance with the phenomenological model of theta phase precession by Yamaguchi et al. [4]. The schematic structure of the hypothesized entorhinal cortex and multimodal sensory system is illustrated in Fig. 4. The entorhinal layer includes head direction cells in the deeper layer and grid cells in the superficial layer. Cells with theta phase precession can be considered as stellate cells. The set of modules along vertical direction form a kind of functional column with a direction preference. These columns form a hypercolumnar structure with a set of directions. Mechanisms in individual modules are explained below.
The local path integration module consists of six units. During animal’s locomotion with a given head direction and velocity, each unit integrates running distance in each direction with an angle dependent coefficient. Units have preferred vector directions distributing with π /3 intervals as shown in Fig. 5. Computation of animal displacement in given directions in this module is illustrated in Fig, 6. The maximum integration length of the distance in each direction is assumed to be common in a module, corresponding to the distance between nearby vertices of the subsequently formed grid field. This computation gives (n.m) in eq. (1). These systems distribute in the deeper layer of the entorhinal cortex in agreement with observation of head direction cells. Different modules have different vector A Computational Model of Formation of Grid Field and Theta Phase Precession 155 directions and form a hypercolumn set covering the entire running directions or entire orientation of resultant grid field. The entorhinal cortex is considered to include multiple hypercolumns with different spatial scales. They are considered to work in parallel possibly to give stability in a global space by compensating accumulation of local errors.
Fig. 5. Left) A module of local path integrator with hexagonal direction vectors and a common vector size. Right) Activity coefficient of each vector unit. A set of these vector units to give a displacement distance measure computes an animal motion in a give head direction.
Fig. 6. Illustration of computation of local path integration in a module. Animal locomotion in a give head direction is computed by a pair of vector units among six vectors to give a position measure.
Computational results of local path integration are projected to next module in the superficial layer of the entorhinal cortex, which has multiple sensory inputs in a given environment. The association of path integration and visual cues results in the relative location of path integration measure ( α
, β
) in eq. (1) in the module. Further interaction in a set of three cells as shown in Fig. 7 can give robustness of the parameter ( α
, β
). Possible interaction among these cells is mutual inhibition to give supplementary distribution of three grid fields. 156 Y. Yamaguchi et al. 2.3 Theta Phase Precession in the Grid Field The input of the parameter (n, m) and ( α
, β i ), to a cell at next module, at the top part of the module, can cause theta phase precession. It is obtained by the fundamental mechanism of theta phase generation proposed by Yamaguchi et al. [4] [5]. The mechanism needs the presence of a gradual increase of natural frequency in a cell with oscillation activity. Here we find that the top module consists of stellate cells with intrinsic theta oscillation. The natural increase in frequency is expected to emerge by the input of path integration at each vertex of a grid field.
Fig. 7. A triplet of grid fields with the same local path integration system and different spatial phases can generate mostly uniform spatial representation where a single grid cell fires at every location. The uniformity can help robust assignment of environmental spatial phases under the help of environmental sensory cues. The association is processed in the middle part of each column in the entorinal cortex. Projection of each cell output to the module at the top generates a grid field wit theta phase precession as explained in text.
Simple mathematical formulation of the above model is phenomenologically given below. The locomotion of animal is represented by current displacement (R,
φ ) computed with head direction φ H , and running velocity. An elementary vector at a column of in local path integration system has a vector angle φ
and its length A . The
output of the i th vector system I is given by
mod
(
) c cos( ) ( with
otherwise,
0 ,
(
nd 2 / H if 1 ) (
S i R i S r i S r a i i I φ φ φ φ π φ φ φ − = ⎪⎩ ⎪ ⎨ ⎧ < < −
− =
A Computational Model of Formation of Grid Field and Theta Phase Precession 157 where r and A respectively represent the field radius and the distance between neighboring grid vertices. The output of path integration module Di to the middle layer is given by
= I ( φ i k ∏ + k π / 3). (3) Through association with visual cues, spatial phase of the grid is determined. (Details are not shown here.) The term Eqs. (2-3) from the middle layer to the top layer gives on-off regulation and also a parameter with gradual increase in a grid field. Dynamics of the membrane potential G i of the cell at the top layer is given by
=
i , t) +
φ
) +
theta ,
(4)
where f is a function of time-dependent ionic currents and a is constant. The last term I theta denotes a sinusoidal current representing theta oscillation of inhibitory neurons. In a proper dynamics of f, the second term in the right had side gives activation of the grid cell oscillation and gradual increase in its natural frequency. According to our former results by using a phenomenological model [5], the last term of theta currents leads phase locking of grid cells with gradual phase shift. This realizes a cell with grid field and theta phase precession. One can test Eq. (4) by applying several types of equations including a simple reduced model and biophysical model of the hippocampus or entorhinal cells. An example of computer experiments is given in the following section. 4 Computer Simulation of Theta Phase Precession The mechanism of theta phase precession was phenomenologically proposed by Yamaguchi et al. [4][5] as coupling of two oscillations. One is LFP theta oscillation with a constant frequency of theta rhythm. The other is a sustained oscillation with gradual increase in natural frequency. The sustained oscillation in the presence of LFP theta exhibits gradual phase shift as quasi steady states of phase locking. The simulation by using a hippocamal pyramidal cell [6] is shown in Fig. 8. It is obviously seen that LFP theta instantaneously captures the oscillation with gradual increase in natural frequency into a quasi-stable phase at each theta cycle to give gradual phase shift. This phase shift is robust against any perturbation as a consequence of phase locking in nonlinear oscillations. The simulation with a model of an entorhinal stellate cell [7] was also elucidated. We obtained similar phase precession with stellate cell model. One important property of stellate cell is the presence of sub threshold oscillations, while synchronization of this oscillation can be reduced to a simple behavior of the phase model. Thus, the mechanism of phenomenological model [5] is found to endow comprehensive description of phase locking of complex biophysical neuron models.
158 Y. Yamaguchi et al.
(a)
(b) Fig. 8. Computer experiment of theta phase precession by using a hippocampal pyramidal neuron model [6]. (a)Bottom: Input current with gradual increase. Top: Resultant sustained oscillation with gradual increase in natural frequency of the membrane potential. (b) In the presence of LFP theta (middle), the neuronal activity exhibit theta phase precession. 5 Discussions and Conclusion We elucidated a computational model of grid cells in the entorhnal cortex to investigate how temporal coding works for spatial representation in the brain, A computational model of formation of grid field was proposed based on local path integration. This assumption was found to give theta phase precession within the grid field. This computational mechanism does not need an assumption of learning in repeated trials in a novel environment but enables instantaneous spatial representation. Furthermore, this model has good agreements with experimental observations of head direction cells and grid cells. The networks proposed in the model predict local interaction networks in the entorhinal cortex and also head direction systems distributed in many areas. Although computation of place cells based on grid cells is beyond this paper, emergence of theta phase precession in the entorhinal cortex can be used for place cell formation and also instantaneous memory formation in the hippocampus [8]. These computational model studies with space-time structure for environmental space representation enlightens the temporal coding over distributed areas used in real-time operation of spatial information in ever changing environment.
A Computational Model of Formation of Grid Field and Theta Phase Precession 159
1. O’Keefe, J., Nadel, L.: The hippocampus as a cognitive map. Clarendon Press, Oxford (1978) 2. Fyhn, M., Molden, S., Witter, M., Moser, E.I., Moser, M.B.: Spatial representation in the entorhinal cortex. Sience 305, 1258–1264 (2004) 3. Hafting, T., Fyhn, M., Moser, M.B., Moser, E.I.: Phase precession and phase locking in entorhinal grid cells. Program No. 68.8, Neuroscience Meeting Planner. Atlanta, GA: Society for Neuroscience (2006.) Online (2006) 4. Yamaguchi, Y., Sato, N., Wagatsuma, H., Wu, Z., Molter, C., Aota, Y.: A unified view of theta-phase coding in the entorhinal-hippocampal system. Current Opinion in Neurobiology 17, 197–204 (2007) 5. Yamaguchi, Y., McNaughton, B.L.: Nonlinear dynamics generating theta phase precession in hippocampal closed circuit and generation of episodic memory. In: Usui, S., Omori, T. (eds.) The Fifth International Conference on Neural Information Processing (ICONIP 1998) and The 1998 Annual Conference of the Japanese Neural Network Society (JNNS 1998), Kitakyushu, Japan. Burke, VA, vol. 2, pp. 781–784. IOS Press, Amsterdam (1998) 6. Pinsky, P.F., Rinzel, J.: Intrinsic and network rhythmogenesis in a reduced traub model for CA3 neurons. Journal of Computational Neuroscience 1, 39–60 (1994) 7. Fransén, E., Alonso, A.A., Dickson, C.T., Magistretti, J., Hasselmo, M.E.: Ionic mechanisms in the generation of subthreshold oscillations and action potential clustering in entorhinal layer II stellate neurons 14(3), 368–384 (2004) 8. Molter, C., Yamaguchi, Y.: Theta phase precession for spatial representation and memory formation. In: The 1st International Conference on Cognitive Neurodynamics (ICCN 2007), Shanghai, 2-09-0002 (2007) Working Memory Dynamics in a Flip-Flop Oscillations Network Model with Milnor Attractor David Colliaux 1,2 , Yoko Yamaguchi 1 , Colin Molter 1 , and Hiroaki Wagatsuma 1 1 Lab for Dynamics of Emergent Intelligence, RIKEN BSI, Wako, Saitama, Japan 2 Ecole Polytechnique (CREA), 75005 Paris, France david.colliaux@polytechnique.org Abstract. A phenomenological model is developed where complex dy- namics are the correlate of spatio-temporal memories. If resting is not a classical fixed point attractor but a Milnor attractor, multiple oscilla- tions appear in the dynamics of a coupled system. This model can be helpful for describing brain activity in terms of well classified dynamics and for implementing human-like real-time computation. 1 Introduction Neuronal collective activities of the brain are widely characterized by oscillations in human and animals [1][2]. Among various frequency bands, distant synchro- nization in theta rhythms (4-8 Hz oscillation defined in human EEG) is recently known to relate with working memory, a short-term memory for central execu- tion in human scalp EEG [3][4] and in neural firing in monkeys [5][6]. For long-term memory, information coding is mediated by synaptic plasticity whereas short-term memory is stored in neural activities [7]. Recent neuroscience reported various types of persistent activities of a single neuron and a population of neurons as possible mechanisms of working memory. Among those, bistable states, up- and down-states, of the membrane potential and its flip-flop tran- sitions were measured in a number of cortical and subcortical neurons. The up-state, characterized by frequent firing, shows stability for seconds or more due to network interactions [8]. However it is little known whether flip-flop tran- sition and distant synchronization work together or what kind of processings are enabled by the flip-flop oscillation network. Associative memory network with flip-flop change was proposed for working memory with classical rate coding view [9], while further consideration on dy- namical linking property based on firing oscillation, such as synchronization of theta rhythms referred above, is likely essential for elucidation of multiple at- tractor systems. Besides, Milnor extended the concept of attractors to invariant sets with Lyapunov unstability, which has been of interest in physical, chemical and biological systems. It might allow high freedom in spontaneous switching among semi-stable states [12]. In this paper, we propose a model of oscillation associative memory with flip-flop change for working memory. We found that M. Ishikawa et al. (Eds.): ICONIP 2007, Part I, LNCS 4984, pp. 160–169, 2008. c Springer-Verlag Berlin Heidelberg 2008 Working Memory Dynamics in a Flip-Flop Oscillations Network Model 161
the Milnor attractor condition is satisfied in the resting state of the model. We will first study how the Milnor attractor appears and will then show possible behaviors of coupled units in the Milnor attractor condition. 2 A Network Model 2.1 Structure In order to realize up- and down-states where up-state is associated with oscil- lation, phenomenological models are joined. Traditionally, associative memory networks are described by state variables representing the membrane potential {S i } [9]. Oscillation is assumed to appear in the up-state as an internal process within each variable φ i
i th unit. Oscillation dynamics is simply given by a phase model with a resting state and periodic motion [10,11]. cos(φ
i ) stands
for an oscillation current in the dynamics of the membrane potential. 2.2
Mathematical Formulation of the Model The flip-flop oscillations network of N units is described by the set of state variables {S i , φ i } ∈ N × [0, 2π[ N (
i and
φ i is given by the following equations: dS i dt = −S i + W ij R(S
j ) +
σ(cos(φ i ) − cos(φ 0 )) + I ± dφ i dt = ω + (β − ρS i ) sin(φ i ) (1) with
R(x) = 1 2 (tanh(10( x − 0.5)) + 1), φ 0 =
−ω β ) and cos(φ 0 ) < 0. R is the spike density of units and input I ±
I + ) or negative ( I − ) pulses (50 time steps), so that we can focus on the persistent activity of units after a phasic input. ω and β are respectively the frequency and the stabilization coefficient of the internal oscillation. ρ and σ represent mutual feedback between internal oscillation and membrane potential. W ij
the connection weights describing the strength of coupling between units i and j. φ 0 is known to be a stable fixed point of the equation for φ, and 0 to be a fixed point for the S equation. 3 An Isolated Unit 3.1 Resting State The resting state is the stable equilibrium when I = 0 for a single unit. We assume ω < β so that M 0 = (0
, φ 0 ) is the fixed point of the system. To study the linear stability of this fixed point, we write the stability matrix around M 0 : DF |
M 0 = −1 −σsin(φ
0 ) −ρsin(φ 0 ) βcos(φ 0 ) (2) 162 D. Colliaux et al. The sign of the eigenvalues of DF |
M 0 and thus the stability of M 0 depends only on μ = ρσ. With our choice of ω = 1 and β = 1.2, μ c ≈ 0.96. If μ < μ c , M 0 is a stable fixed point and there is another fixed point M 1 = ( S 1 , φ 1 ) with
φ 1
0 which is unstable. If μ > μ c , M 0 is unstable and M 1 is stable with φ 1 > φ 0 . Fixed
points exchange stability as the bifurcation parameter μ increases (transcritical bifurcation). The simplified system according to eigenvectors ( X 1 , X 2 ) of the matrix DF |
M 0 gives a clear illustration of the bifurcation as dx 1 dt = ax 2 1 + λ 1 x 1 dx 2 dt = λ 2 x 2 (3) Here a = 0 is equivalent to μ = μ c and in this condition there is a positive measure basin of attraction but some directions are unstable. The resting state M 0 is not a classical fixed point attractor because it does not attract all tra- jectories from an open neighborhood, but it is still an attractor if we consider Milnor’s extended definition of attractors. Phase plane ( S, φ) Fig. 1 shows that for μ close to the critical value, nullclines cross twice staying close to each other in between. That narrow channel makes the configuration indistinguishable from a Milnor attractor in computer experiments. Fig. 1. Top: Phase space (S, φ) with vector field and nullclines of the system. The dashed domain in B shows that M 0
μ = μ c . Bottom: Fixed points with their stable and unstable directions for the equivalent simplified system. A: μ < μ c
μ = μ c . C: μ > μ c . Since we showed μ is the crucial parameter for the stability of the resting state, we can now consider ρ = 1 and study the dynamics according to σ with a close look near the critical regime ( σ = μ
c ).
Working Memory Dynamics in a Flip-Flop Oscillations Network Model 163
3.2 Constant Input Can Give Oscillations Under constant input there are two possible dynamics: fixed point and limit cycle. If ω β − S
< 1 (4)
there is a stable fixed point ( S 1 , φ 1 ) with φ 1 solution of ω + (β − σ(cos(φ 1 ) − cos(φ 0 )) − I)sin(φ 1 ) = 0 S 1 = σ(cosφ 1 − cosφ 0 ) +
I (5)
If condition 4 is not satisfied, the φ equation in 1 will give rise to oscillatory dynamics. Identifying S with its temporal average, dφ dt = ω + Γ sin(φ) with Γ = β − S will be periodic with period 2π 0 dφ ω+(β−S)sin(φ) . This approximation gives an oscillation at frequency ω = ω 2
2 , which is qualitatively in good agreement with computer experiments Fig. 2. -1 0 1 2 3 4 5 6 -0.4 -0.2
0 0.2
0.4 0.6
0.8 1 1.2 1.4 0 0.2 0.4 0.6
0.8 1 1.2 S f I σ = μ c S minimum S maximum Frequency (theoretical) Frequency Fig. 2. For each value of constant current I, maximum and minimum values of S 1 are
plotted. Dominant frequency of S 1 obtained by FFT is compared to the theoretical value when S is identified with its temporal average: Frequency VS Frequency (theo- retical). If we inject an oscillatory input into the system, S oscillates at the same frequency provided the input frequency is low. For higher frequencies, S can- not follow the input and shows complex oscillatory dynamics with multiple frequencies.
164 D. Colliaux et al. 4 Two Coupled Units For two coupled units, flip-flop of oscillations is observed under various con- ditions. We will analyze the case μ = 0 and flip-flop properties under vari- ous strengths of connection weights, assuming symmetrical connections ( W 12
W 2,1
= W ).
4.1 Influence of the Feedback Loop In equation 1, ρ and σ implement a feedback loop representing mutual influence of φ and S for each unit. The Case µ = 0. In the case σ = 0 or ρ = 0, φ remains constant φ = φ 0 :
associative memory network storing patterns in fixed point attractors [9]. For small coupling strength, the resting state is a fixed point. For strong coupling strength, two more fixed points appear, one unstable, corresponding to threshold, and one stable, providing memory storage. After a transient positive input I +
input I − can bring it back to resting state. For a small perturbation ( σ 1 and
ρ = 1), the active state is a small up-state oscillation but associative memory properties (storage, completion) are preserved. Growing Oscillations. The up-state oscillation in the membrane potential dynamics triggered by giving an I + pulse to unit 1 grows when σ increases and saturates to an up-state fixed point for strong feedback. Interestingly, for a range of feedback strength values near μ c
attractor resting state. Projection of the trajectories of the 4-dimensional system on a 2-dimensional plane section P illustrates these complex dynamics Fig. 3. A cycle would intersect this plane in two points. For each σ value, we consider S 1 for these intersection points. For a range between 0.91 and 1.05 with our choice of parameters, there are much more than two intersection points M*, suggesting chaotic dynamics. 4.2 Influence of the Coupling Strength The dynamics of two coupled units can be a fixed point attractor, as in the resting state ( I = 0), or down-state or up-state oscillation (depending on the coupling strength), after a transient input. Near critical value of the feedback loop, in addition to these, more complex dynamics occur for intermediate cou- pling strength. Working Memory Dynamics in a Flip-Flop Oscillations Network Model 165
Fig. 3. A: Influence of the feedback loop- Bifurcation diagram according to σ (Top). S 1 coordinates of the intersecting points of the trajectory with a plane section P according to σ(Bottom). B: Influence of the coupling strengh - S 1 maximum and minimum values and average phase difference ( φ 1 − φ 2 ) according to W (Top). S 1 coordinates of the intersecting points of the trajectory with a plane section P according to W (Bottom). Down-state Oscillation. For small coupling strength, the system periodically visits the resting state for a long time and goes briefly to up-state. The frequency of this oscillation increases with coupling strength. The two units are anti-phase (when S
takes maximum value, S j takes minimum value) Fig. 4 (Bottom). Up-state Oscillation. For strong coupling strength, a transient input to unit 1 leads to an up-state oscillation Fig. 4 (Top). The two units are perfectly in-phase at W = 0.75 and phase difference stays small for stronger coupling strength. Chaotic Dynamics. For intermediate coupling strength, an intermediate cycle is observed and more complex dynamics occur for a small range (0 .58 166 D. Colliaux et al. Fig. 4. S i temporal evolution, ( S 1 , S 2 ) phase plane and ( S i
i ) cylinder space. Top: Up-state oscillation for strong coupling. Middle: Multiple frequency oscillation for in- termediate coupling. Bottom: Down-state oscillation for weak coupling. with our parameters) before full synchronization characterized by φ 1 − φ 2 = 0. The trajectory can have many intersection points with P and S ∗ in Fig. 3 shows multiple roads to chaos through period doubling. Working Memory Dynamics in a Flip-Flop Oscillations Network Model 167
5 Application to Slow Selection of a Memorized Pattern 5.1 A Small Network The network is a set N of five units consisting in a subset N 1 of three units A,B and C and another N 2
metrical all-to-all weak connections ( W N = 0 .01) and in each subset units have symmetrical all-to-all strong connections ( W N i = 0
.1 ∗ M) with M a global pa- rameter slowly varying in time between 1 and 10. These subsets could represent two objects stored in the weight matrix. 5.2
Memory Retrieval and Response Selection We consider a transient structured input into the network. For constant M, a partial or complete stimulation of a subset N i can elicit retrieval and completion of the subset in an up-state as would do a classical auto-associative memory network. 0 0.5 1 1.5
2 0 20000 40000 60000
80000 100000
120000 140000
S t A 0 0.5
1 1.5
2 0 20000 40000 60000
80000 100000
120000 140000
S t B 0 0.5
1 1.5
2 0 20000 40000 60000
80000 100000
120000 140000
S t C 0 0.5
1 1.5
2 0 20000 40000 60000
80000 100000
120000 140000
S t D 0 0.5
1 1.5
2 0 20000 40000 60000
80000 100000
120000 140000
S t E Fig. 5. Slow activation of a robust synchronous up-state in N 1 during slow increase of M In the Milnor attractor condition more complex retrieval can be achieved when M is slowly increased. As an illustration, we consider transient stimulation of units A and B from N 1
N 2 Fig. 5. N 2 units show anti-phase 168 D. Colliaux et al. oscillations with increasing frequency. N 1 units first show synchronous down- state oscillations with long stays near the Milnor attractor and gradually go toward sustained up-state oscillations. In this example, the selection of N 1 in up-state is very slow and synchrony between units plays an important role. 6 Conclusion We demonstrated that, in cylinder space, a Milnor attractor appears at a crit- ical condition through forward and reverse saddle-node bifurcations. Near the critical condition, the pair of saddle and node constructs a pseudo-attractor, which can serves for observation of Milnor attractor-like properties in computer experiments. Semi-stability of the Milnor attractor in this model seems to be associated with the variety of oscillations and chaotic dynamics through period doubling roads. We demonstrated that an oscillations network provides a variety of working memory encoding in dynamical states under the presence of a Milnor attractor. Applications of oscillatory dynamics have been compared to classical autoas- sociative memory models. The importance of Milnor attractors was proposed in the analysis of coupled map lattices in high dimension [11] and for chaotic itinerancy in the brain [13]. The functional significance of flip-flop oscillations networks with the above dynamical complexity is of interest for further analysis of integrative brain dynamics. References 1. Varela, F., Lachaux, J.-P., Rodriguez, E., Martinerie, J.: The brainweb: Phase synchronization and large-scale integration. Nature Reviews Neuroscience (2001) 2. Buzsaki, G., Draguhn, A.: Neuronal oscillations in cortical networks. Science (2004) 3. Onton, J., Delorme, A., Makeig, S.: Frontal midline EEG dynamics during working memory. NeuroImage (2005) 4. Mizuhara, H., Yamaguchi, Y.: Human cortical circuits for central executive function emerge by theta phase synchronization. NeuroImage (2004) 5. Rainer, G., Lee, H., Simpson, G.V., Logothetis, N.K.: Working-memory related theta (4-7Hz) frequency oscillations observed in monkey extrastriate visual cortex. Neurocomputing (2004) 6. Tsujimoto, T., Shimazu, H., Isomura, Y., Sasaki, K.: Prefrontal theta oscillations associated with hand movements triggered by warning and imperative stimuli in the monkey. Neuroscience Letters (2003) 7. Goldman-Rakic, P.S.: Cellular basis of working memory. Neuron (1995) 8. McCormick, D.A.: Neuronal Networks: Flip-Flops in the Brain. Current Biology (2005)
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