Formation of working memory in a spiking neuron network accompanied by astrocytes
Susanna Yu. Gordleeva, Yulia A. Tsybina, Mikhail I. Krivonosov, Mikhail V. Ivanchenko, Alexey A. Zaikin, Victor B. Kazantsev, Alexander N. Gorban
FFormation of working memory in a spiking neuron networkaccompanied by astrocytes
Susanna Yu. Gordleeva a,b, ∗ , Yulia A. Tsybina a , Mikhail I. Krivonosov a , Mikhail V. Ivanchenko a ,Alexey A. Zaikin a,c,d , Victor B. Kazantsev a,b , Alexander N. Gorban a,e a Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia b Neuroscience and Cognitive Technology Laboratory, Center for Technologies in Robotics and MechatronicsComponents, Innopolis University, Innopolis, Russia c Centre for Analysis of Complex Systems, Sechenov First Moscow State Medical University (Sechenov University),Moscow, Russia d Institute for Women’s Health and Department of Mathematics, University College London, London, UK e University of Leicester, UK
Abstract
We propose a biologically plausible computational model of working memory (WM) imple-mented by the spiking neuron network (SNN) interacting with a network of astrocytes. SNN ismodelled by the synaptically coupled Izhikevich neurons with a non-specific architecture con-nection topology. Astrocytes generating calcium signals are connected by local gap junctiondi ff usive couplings and interact with neurons by chemicals di ff used in the extracellular space.Calcium elevations occur in response to the increase of concentration of a neurotransmitter re-leased by spiking neurons when a group of them fire coherently. In turn, gliotransmitters arereleased by activated astrocytes modulating the strengths of synaptic connections in the corre-sponding neuronal group. Input information is encoded as two-dimensional patterns of shortapplied current pulses stimulating neurons. The output is taken from frequencies of transientdischarges of corresponding neurons. We show how a set of information patterns with quitesignificant overlapping areas can be uploaded into the neuron-astrocyte network and stored forseveral seconds. Information retrieval is organised by the application of a cue pattern represent-ing the one from the memory set distorted by noise. We found that successful retrieval withlevel of the correlation between recalled pattern and ideal pattern more than 90% is possible formulti-item WM task. Having analysed the dynamical mechanism of WM formation, we dis-covered that astrocytes operating at a time scale of a dozen of seconds can successfully storetraces of neuronal activations corresponding to information patterns. In the retrieval stage, theastrocytic network selectively modulates synaptic connections in SNN leading to the success-ful recall. Information and dynamical characteristics of the proposed WM model agrees withclassical concepts and other WM models. Keywords: spiking neural network, astrocyte, neuron-astrocyte interaction, working memory ∗ Corresponding author
Preprint submitted to Elsevier November 4, 2020 a r X i v : . [ q - b i o . N C ] N ov . Introduction In neuroscience, the understanding of the functional role of astrocytes in CNS is still open todebate (Savtchouk and Volterra, 2018), but now there is much evidence demonstrating the in-volvement of astrocytes in local synaptic plasticity and coordination of network activity (Durkeeand Araque, 2019), and as a result in information processing and memory encoding (Santelloet al., 2019). Astrocytes sense synaptic activity and respond to it with the transient elevation ofthe intracellular Ca + concentration (lasting from hundreds of a millisecond to a dozen of sec-onds). Such Ca + signals in astrocytes have been observed in di ff erent brain regions and alsoin the cortex, appearing there in response to the mechanic sensory stimulation (Stobart et al.,2018; Wang et al., 2006a; Takata et al., 2011) and visual sensory stimulation (Chen et al., 2012;Schummers et al., 2008; Perea et al., 2014). Ca + activation can trigger the release of gliotrans-mitters from astrocyte, which in turn a ff ect the dynamics of presynaptic and postsynaptic termi-nals resulting in modulations of synaptic transmission (Araque et al., 2014). The gliotransmitter-mediated synaptic modulation lasts from a dozen of seconds (Jourdain et al., 2007; Perea et al.,2014) to a dozen of minutes (Perea and Araque, 2007; Navarrete et al., 2012; Stellwagen andMalenka, 2006) contributing to both short- and long-term synaptic plasticity. Obviously, thereis a qualitative coincidence of time scales of astrocyte-mediated synaptic modulation with theworking memory (WM) timings during decision making. Based on this and the other followingfacts of astrocyte participation in neuronal signalling, we hypothesized that the astrocytes maybe involved in the WM formation. In particular, recent in vivo studies have shown the partici-pation of astrocytes in synchronization of certain cortical network activity (Takata et al., 2011;Chen et al., 2012; Perea et al., 2014; Paukert et al., 2014), cognitive functions and behaviour(Sardinha et al., 2017; Poskanzer and Yuste, 2016). Experimental evidence shows that astrocytepathology in medium PFC impairs the WM and learning functions (Lima et al., 2014), increasingof astrocyte density enhances short-term memory performance (Luca et al., 2020), recognitionmemory performance and disruption of WM depend on the gliotransmitter release from astrocytein the hippocampus (Robin et al., 2018; Han et al., 2012). Despite these numerous experimen-tal insights of the contribution of astrocytes to synaptic modulations in neuronal signalling, thepossible role of astrocyte in information processing and learning is still a subject of discussion(Kastanenka et al., 2019; Kanakov et al., 2019).Considering the significance of WM processes, the challenge of finding alternative mecha-nisms and the experimental evidence of the astrocytic role in information processing in CNS, itis interesting to study astrocyte-induced modulation of synaptic transmission in the WM organ-isation. Specifically, we assume that the NMDAR-mediated potentiation of excitatory synapsesinduced by the D-serine released from astrocyte in PFC (Fossat et al., 2011; Takata et al., 2011;Chen et al., 2012) plays an essential role in WM. To test this hypothesis, we developed a novelneuron-astrocyte network model for visual WM to reflect experimental data on the structure, con-nectivity, and neurophysiology of the neuron-astrocytic interaction in underlying cortical tissue.We focused on the implementation of a multi-item WM task in DMS framework representingclassical neuropsychological paradigm (Miller et al., 1996), which was previously studied usinga recurrent neural network (Brunel and Wang, 2001; Amit, 2003; Amit et al., 2013; Fiebig andLansner, 2016). In our model, memory is associated with item-specific patterns of astrocyte-induced enhancement of excitatory synaptic transmission. We show how the biologically rele-vant neuron-astrocyte network model implements loading, storage and cued retrieval of multipleitems with significant overlapping. The memory items are encoded in neuronal populations inthe form of discrete high-frequency bursts rather than persistent spiking.2n the following, we review some related works (Section 2), describe the proposed model andmethods in detail (Section 3), present the results (Section 4), and finally, we conclude this workin Section 5.
2. Related work
The concept of WM proposes the ability to temporarily store and process information in goal-directed behaviour. WM is crucial in the generation of higher cognitive functions for both hu-mans and other animals (Baddeley, 1986; Conway et al., 2003; Baddeley, 2012). In primates,visual WM has been studied in delay tasks, such as delayed matching to sample (DMS), whichrequire memory to be held during a brief delay period lasting for several seconds (Miller et al.,1996). Recordings in the monkeys’ prefrontal cortices (PFCs) during the delay task showedthat some neurons displayed persistent and stimulus-specific delay-period activity (Fuster andAlexander, 1971; Funahashi et al., 1989; Barak et al., 2010; Shafi et al., 2007; Funahashi, 2017).Delay persistent activity is considered the neural correlate of WM (Goldman-Rakic, 1995; Con-stantinidis et al., 2018).The classical theoretical memory models suggest that an information item can be stored withsustained neural activity which emerges via activation of stable activity patterns in the network(e.g. attractors) (Hopfield, 1982; DJ, 1995; Wang, 2001; Wimmer et al., 2014) recently reviewedby Zylberberg and Strowbridge (2017); Chaudhuri and Fiete (2016). These WM models proposethat the generation of the persistent activity can be the result of an intrinsic property of theneurons (including the generation of the bistability mediated by the voltage-gated inward currents(Kass and Mintz, 2005) and Ca + -triggered long-term changes in neuronal excitability (Frans´enet al., 2006)) and can be induced by the connectivity within the neural circuit with feed-forward(Ganguli and Latham, 2009; Goldman, 2009) or recurrent (Kilpatrick et al., 2013; Koulakovet al., 2002) architecture. In such models, memory recall is impossible from a silent inactive state.For many WM models of persistent activity based on recurrent connectivity, small deviations inthe network structure destroy the persistence. Moreover, a spiking form of storage information isenergetically unfavourable because of the high metabolic value of action potentials (Attwell andLaughlin, 2001).In theoretical studies a concept of oscillatory sub-cycles storing 7 ± ff erent phases of rhythmic oscillations (Klinshov and Nekorkin, 2008; Borisyuk et al., 2013).Recently, the persistent activity hypothesis has been undergoing critical reviews (Lundqvistet al., 2018) based on the experimental findings in the rodents, rats, and primates showing thatthe robust persistent activity does not last for the entire delay period, but rather sequential neu-ronal firing is observed during the delay period suggesting that the PFC neural network may sup-port WM based on dynamically-changing neuronal activity (Fujisawa et al., 2008; Runyan et al.,2017; Park et al., 2019; Ozdemir et al., 2020; Lundqvist et al., 2016). Despite the considerableprogress which has been made in identifying the neurophysiological mechanisms contributingto WM in mammals (D’Esposito and Postle, 2015; Zylberberg and Strowbridge, 2017), the on-going debate focuses on the generation mechanisms of the delay period activity that appears tounderlie the WM (Constantinidis et al., 2018; Sreenivasan and D’Esposito, 2019). Currently, oneof the recognised experimentally based hypotheses of the WM mechanism underlining the delayactivity (not necessarily persistent) is the synaptic plasticity in the PFC (Tsodyks and Markram,1997; Wang et al., 2006b; Hempel et al., 2000; Erickson et al., 2010). Synaptic plasticity implies3 rapid regulation of the strengths of individual synapses in response to specific patterns of cor-related synaptic activity and contributes to the activity-dependent refinement of neural circuitry.Following these findings, alternative synaptic-based WM models have been proposed (Mongilloet al., 2008; Manohar et al., 2019; Barak and Tsodyks, 2014; Koutsikou et al., 2018). In thesemodels, memory items are stored by stimulus-specific patterns of synaptic facilitation in neuronalcircuit. Synaptic plasticity does not require neurons to show a persistent activity for the entireperiod of the memory task, which results in a robust and more metabolically e ffi cient mechanism.Some synaptic WM models based on short-term non-associative synaptic facilitation (Mongilloet al., 2008; Lundqvist et al., 2011; Mi et al., 2017) allow reading out and refreshing existingrepresentations maintained in the synaptic structure. Others have proposed fast Hebbian activity-dependent synaptic plasticity (Fiebig and Lansner, 2016; Sandberg et al., 2003) for encoding andmaintenance of novel associations.There are a few attempts to investigate theoretically the role of astrocyte-induced modulationof synaptic transmission in memory formation. Shen and Wilde (2007) demonstrate one of thefirst results of simulating the coupling of Hopfield neural network, astrocyte, and cerebrovascularactivity. Although there remained a far stretch to a biophysical model, the resulted suggestedthat a modification of the synapse strengths allows the neuronal firing and the cerebrovascularflow to be compatible on a meso-scale; with astrocyte signalling added, limit cycles exist in thecoupled networks. Tewari and Parpura (2013) and Wade et al. (2011) study how bidirectionalcoupling between astrocytes and neurons in small neuron-astrocyte ensembles mediates learningand dynamic coordination in the brain. Recent interesting theoretical study proposes a self-repairing spiking astrocyte-neural network combined with novel learning rule based on the spike-timing-dependent plasticity and Bienenstock, Cooper, and Munro learning rule (Liu et al., 2019).
3. Materials and methods
Even though the balance of inhibition and excitation play was shown to play an important rolein WM stabilization and to influence the WM capacity (Barak and Tsodyks, 2014), we focus onthe properties of astrocyte-induced modulation of excitatory synaptic transmission in the PFC.We take spiking neuronal network with dimension W × H consisting of synaptically coupled ex-citatory neurons formed by the Izhikevich model (Izhikevich, 2003). Neurons in the network areconnected randomly with the connection length determined by the exponential distribution. It hasbeen experimentally estimated that there is a little overlap in the spatial territories occupied byindividual astrocytes in the cortex (Halassa et al., 2007). An individual cortical astrocyte contactson average 4-8 neuronal somata and 300–600 neuronal dendrites (Halassa et al., 2007). A corticalastrocyte has a “bushy” ramified structure of the fine perisynaptic processes, which cover mostof neuronal membranes within their reach (Allen and Eroglu, 2017). This allows the astrocyte tointegrate and coordinate a unique volume of synaptic activity. Following the experimental data,astrocytic network compartment of our model is organised as a two-dimensional square latticewith only nearest-neighbour connectivity. Each astrocyte interacts with the neuronal ensembleof N a neurons with small overlapping. We consider the bidirectional communication betweenneuronal and astrocytic networks. The scheme of the network topology is shown in Fig. 1.Model equations are integrated using the Runge-Kutta fourth-order method with a fixed timestep, ∆ t = . n p u t s i g n a l N e u r o n a l n e t w o r k A s t r o c y t i c n e t w o r k Figure 1: Neuron-astrocyte network topology. Neuron-astrocyte network consists of two interacting layers: neuralnetwork layer and astrocytic layer. Neuronal network with dimension W × H (79 ×
79) consists of synaptically coupledexcitatory neurons modelled by the Izhikevich neuron. Neurons in network are connected randomly. The astrocyticnetwork consists of di ff usely connected astrocytes with dimension M × N (26 × N a =
16 neurons with dimensions 4 × action parameters) and 4 (stimulation and recall testing). The code is available at the linkhttps: // github.com / altergot / neuro-astro-network. Dynamics of the membrane potential of a single neuron is described by the Izhikevich model(Izhikevich, 2003): dV ( i , j ) dt = . V ( i , j ) (2) + V ( i , j ) − U ( i , j ) + + I ( i , j )app + I ( i , j )syn ; dU ( i , j ) dt = a ( bV ( i , j ) − U ( i , j ) ); (1)with the auxiliary after-spike resettingif V ( i , j ) ≥
30 mV, then V ( i , j ) ← cU ( i , j ) ← U ( i , j ) + d , (2)where the superscripts ( i = , . . . , j = , . . . ,
79) correspond to a neuronal index, the trans-membrane potential V is given in mV and time t in ms. The applied currents I ( i , j )app simulate theinput signal. Neurons receive a number of synaptic currents from other presynaptic neuronsin the network, N ( i , j ) in , which are summed at the membrane according to the following equation5 able 1: Neural network parameters (Izhikevich (2003); Kazantsev and Asatryan (2011)) Parameter Parameter description Value W × H neural network grid size 79 × a time scale of the recovery variable 0.1 b sensitivity of the recovery variable to the sub-threshold fluctu-ations of the membrane potential 0.2 c after-spike reset value of the membrane potential -65 mV d after-spike reset value of the recovery variable 2 η synaptic weight without astrocytic influence 0.025 E syn synaptic reversal potential for excitatory synapses ) 0 mV k syn slope of the synaptic activation function 0.2 mV N out number of output connections per each neuron 40 λ rate of the exponential distribution of synaptic connections dis-tance 5(Kazantsev and Asatryan, 2011; Esir et al., 2018): I ( i , j )syn = N ( i , j ) in (cid:88) k = g ( i , j )syn ( E syn − V ( i , j ) )1 + exp( − V k pre k syn ) ; (3)Parameter g ( i , j )syn describes the synaptic weight, g ( i , j )syn = η + ν ( m , n ) Ca . The astrocyte ( m , n ) modulatesthe synaptic currents of the neuron ( i , j ). The variable ν Ca introduces the astrocyte-induced mod-ulation of synaptic strength and will be discussed below. Here, purely for illustration of astrocytee ff ects, we did not include intrinsic short-term synaptic plasticity in the model. The synapticreversal potential for excitatory synapses is taken with E syn = V pre denotes the membranepotential of the presynaptic neuron. For simplicity, we neglect the axonal and synaptic delays.The architecture of synaptic connections between neurons is non-specific (e.g. random) withthe following parameters. The number of output connections per each neuron is fixed at N out =
40. Each neuron innervates N out local postsynaptic targets, the distance R to which is determinedaccording to the exponential distribution: f R ( R ) = λ exp( − λ R ) , R ≥ , , R < . (4) In the model we try to implement the biologically plausible organisation of astrocytic net-work and neuron-astrocyte interaction. The astrocytic network is configured in the form of atwo-dimensional square lattice with dimension M × N . Cortical astrocytes are coupled via CX43gap junctions mostly permeable to inositol 1,4,5-trisphosphate (IP ). Hence, in the model, weconsider local di ff usive coupling. Besides, each astrocyte is interconnected with neuronal en-semble of N a neurons. It was experimentally shown that the sensory stimulation evokes fastintracellular Ca + signals in fine processes of cortical astrocyte in response to local synaptic ac-tivity in neuronal circuit (Stobart et al., 2018; Takata et al., 2011; Wang et al., 2006a). Multiplerapid, spatially restricted Ca + events in the astrocytic process are induced by intense neuronal6ring. Local events are spatially and temporally integrated by the astrocytic cell, which resultsin a global, long lasting Ca + event. In turn, this event induces the release of gliotransmittersa ff ecting synaptic transmission in the local territory of individual astrocytes (Henneberger et al.,2010; Bekar et al., 2008; Araque et al., 2014). For simplicity, we did not model detailed pro-cess of spatial-temporal integration of the rapid Ca + signals in the morphological structure ofastrocyte modelled earlier by Gordleeva et al. (2019, 2018) and Wu et al. (2018). Here we em-ploy a mean-field approach to describe the emergence of a global Ca + signal and its impact onsynchronisation of neuronal ensemble controlled by a certain astrocyte.As pyramidal neurons generate the spike, glutamate is released from the presynaptic terminalinto the synaptic cleft (Fig. 2b,d). Amount of glutamate, G , that di ff used from synaptic cleftand reached the astrocytic process can be described by the following equation (Gordleeva et al.,2012; Pankratova et al., 2019): dG ( i , j ) dt = − α glu G ( i , j ) + k glu Θ ( V ( i , j ) − mV ) , (5)here α glu is the glutamate clearance constant, k glu is the e ffi cacy of the release, Θ denotes theHeaviside step function and V ( i , j ) is the membrane potential of the corresponding presynapticneuron ( i , j ). Binding of glutamate to metabotropic glutamate receptors (mGluR) on the astro-cytic membrane, which is located close to the synapse, triggers the production of IP in theastrocyte (Fig. 2e). We use the approaches from earlier studies to describe the dynamics of theintracellular concentration of IP in the astrocyte (Nadkarni and Jung, 2003; Ullah et al., 2006): dIP ( m , n )3 dt = IP ∗ − IP ( m , n )3 τ IP + J ( m , n )PLC δ + J ( m , n )glu + di ff ( m , n ) IP , (6)with m = , . . . , n = , . . . ,
26. Parameter IP ∗ denotes the steady state concentration of theIP and J PLC δ describes the IP production by phospholipase C δ (PLC δ ) (Ullah et al. (2006)): J PLC δ = v ( Ca + (1 − α ) k ) Ca + k (7)The variable J glu describes the glutamate-induced production of the IP in response to neuronalactivity and is modelled as a rectangular-shaped pulse with amplitude A glu µ M and duration t glu ms: J glu = A glu , if t < t ≤ t + t glu , , otherwise; (8)here t denotes the periods when the total level of glutamate in all synapses associated with thisastrocyte reaches a threshold: N a (cid:88) ( i , j ) ∈ N a [ G ( i , j ) > G thr ] > F act , (9)here we use the parameter G thr = .
7. [] denotes the Iverson bracket. F act is the fraction ofsynchronously spiking neurons of the neuronal ensemble corresponding to the astrocyte. Forthe emergence of the calcium elevation F act = . J glu , is activated when correlated activity in the neuronal ensemble reaches a certain levelof coherence. 7ncrease of IP concentration in the astrocyte induces the release of Ca + from internal stores,mostly from the endoplasmic reticulum (ER), to cytosol. For simplified description of the bio-physical mechanism underlying the calcium dynamics in astrocytes, we use the Ullah model(Ullah et al., 2006). Changes of the intracellular Ca + concentration, Ca , are described by thefollowing equations: dCa ( m , n ) dt = J ( m , n )ER − J ( m , n )pump + J ( m , n )leak + J ( m , n )in − J ( m , n )out + di ff ( m , n ) Ca ; dh ( m , n ) dt = a d IP ( m , n )3 + d IP ( m , n )3 + d (1 − h m , n ) − Ca ( m , n ) h ( m , n ) ; (10)where h is the fraction of the activated IP receptors (IP Rs) on the ER surface. Flux J ER isCa + flux from the ER to the cytosol through IP Rs, J pump is the flux pumped Ca + back intoER via the sarco / ER Ca + -ATPase (SERCA), J leak is the leakage flux from the ER to the cytosol.Fluxes J in and J out describe calcium exchange with extracellular space. The fluxes are expressedas follows: J ER = c v Ca h IP ( c / c − (1 + / c ) Ca )(( IP + d )( IP + d )) ; J pump = v Ca k + Ca ; J leak = c v ( c / c − (1 + / c ) Ca ); J in = v IP k + IP ; J out = k Ca ; (11)Biophysical meaning of all parameters in Eqs. (6), (7),(10),(11) and their values determinedexperimentally can be found in (Li and Rinzel, 1994; Ullah et al., 2006) and in Table 2.Cortical astrocytes are coupled by CX43 gap junctions (Nimmerjahn et al., 2004). Thus,di ff usion of active chemicals becomes possible between the neighbouring astrocytes. Currentsdi ff Ca and di ff IP describe the di ff usion of Ca + ions and IP molecules via gap junctions betweenthe astrocytes in the network and can be expressed as follows:di ff ( m , n ) Ca = d Ca ( ∆ Ca ) ( m , n ) ;di ff ( m , n ) IP = d IP ( ∆ IP ) ( m , n ) ; (12)where parameters d Ca and d IP describe the Ca + and IP di ff usion rates, respectively. Followingexperimental data, we assume that CX43 is less permeable to Ca + than to d IP . ( ∆ Ca ) ( m , n ) and( ∆ IP ) ( m , n ) are the discrete Laplace operators:( ∆ Ca ) ( m , n ) = ( Ca ( m + , n ) + Ca ( m − , n ) + Ca ( m , n + + Ca ( m , n − − Ca ( m , n ) ) . (13)Equations (6)-(8),(10)-(12) predict that the synchronized activity in the neuronal ensemble trig-gers the astrocytic Ca + signals, and in the absence of neuronal stimulus in the astrocytic networksteady state Ca + concentration is maintained.Next, we account for the e ff ect of enhancement of excitatory postsynaptic currents ( ePS C )generation through modulation of postsynaptic NMDARs by D-serine released from the astrocyte8 able 2: Astrocytic network parameters (Ullah et al., 2006) Parameter Parameter description Value M × N astrocytic network grid size 26 × c total Ca + in terms of cytosolic vol 2.0 µ M c (ER vol) / (cytosolic vol) 0.185 v max Ca + channel flux 6 s − v Ca + leak flux constant 0.11 s − v max Ca + uptake 2.2 µ M s − v maximal rate of activation dependent calcium influx 0.2 µ M s − k rate constant of calcium extrusion 0.5 s − k half-saturation constant for agonist-dependent calciumentry 1 µ M k activation constant for ATP-Ca + pump 0.1 µ M d dissociation constant for IP µ M d dissociation constant for Ca + inhibition 1.049 µ M d receptor dissociation constant for IP d Ca + activation constant 82 nM α v max rate of IP production 0.3 µ M s − / τ r rate constant for loss of IP − IP ∗ steady state concentration of IP µ M k dissociation constant for Ca + stimulation of IP produc-tion 1.1 µ M d Ca Ca + di ff usion rate 0.05 s − d IP IP di ff usion rate 0.1 s − (Bergersen et al., 2011; Henneberger et al., 2010). In the model, we propose that global events ofCa + elevation in astrocyte result in D-serine release, which can modulate the synaptic strengthof all synapses corresponding to the morphological territory of this astrocyte. For simplicity,the relationship between the astrocyte Ca + concentration and synaptic weight of the a ff ectedsynapses g syn , is described as follows: g syn = η + ν Ca ,ν Ca = ν ∗ Ca Θ ( Ca ( m , n ) − Ca thr ) (14)where the parameter ν ∗ Ca denotes the strength of the astrocyte-induced modulation of the synap-tic weight, Θ ( x ) is the Heaviside step-function. The feedback from the astrocytes to the neuronsis activated when the astrocytic Ca + concentration is larger than Ca thr and the fraction of syn-chronously spiking neurons of neuronal ensemble corresponding to the astrocyte F astro duringthe time period of τ syn =
10 ms. The duration of the feedback is fixed and is equal to τ astro = The term I app in Eq. (1) represents specific and non-specific external inputs. A non-specificnoisy input simulates the input signals from networks of other brain areas and is applied contin-uously to all neurons in the form of independent Poisson pulse trains of a certain rate, f bg , with9 able 3: Neuron-astrocytic interaction parameters (Gordleeva et al., 2012) Parameter Parameter description Value N a number of neurons interacting with one astrocyte 16, 4 × α glu glutamate clearance constant 10 s − k glu e ffi cacy of the glutamate release 600 µ M s − A glu rate of IP production through glutamate 5 µ M s − t glu duration of IP production through glutamate 60 ms G thr threshold concentration of glutamate for IP production 0.1 F act fraction of synchronously spiking neurons required for theemergence of Ca + elevation 0.5 F astro fraction of synchronously spiking neurons required for theemergence of astrocytic modulation of synaptic transmis-sion 0.375 ν ∗ Ca strength of the astrocyte-induced modulation of synapticweight 0.5 Ca thr threshold concentration of Ca + for the astrocytic modu-lation of synapse 0.15 µ M τ astro duration of the astrocyte-induced modulation of synapse 250 ms Table 4: Stimulation protocol and recall testing parameters
Parameter Parameter description Value f bg background activity rate 1.5 Hz A stim stimulation amplitude 10 µ A t stim stimulation duration 200 msnoise level in sample 5% A test cue stimulation amplitude 8 µ A t test cue stimulation length 150 msnoise level in cue 20%amplitudes randomly and uniformly distributed in the interval [ − , µ A. This input evokesbackground network state with low-rate, spontaneous spiking.Specific input contains training samples in the form binary spatial patterns. The patterns rep-resent di ff erent spatial distributions relative to background state with non-specific input only.Average size of sample is 1078 neurons (18% of network) stimulated by the specific input, withaverage 35 .
2% overlapping in the population. For visual representation of samples, we take bi-nary images of numerals (0,1,2,3,4,..) with size W × H pixels, where each pixel correspondsto a neuron in the neuronal layer. Neurons corresponding to the shape of the numerals receiverectangular excitatory pulse with length t stim and amplitude A stim . The shape of each sample wasspatially distorted by 5% random noise such as “salt and pepper noise”. Then a transient in-puts were applied to simulate the nonmatching test items and the cue (length t test , and amplitude A test ). In the cued recall for simulating the loss in saliency, we applied shorter input with loweramplitude and higher level (20%) of random noise.10 .5. Memory performance metrics To measure memory performance of the system we count the correlation of recalled patternwith ideal item in the following way: M i j ( t ) = I t (cid:88) k = t − w I [ V i j ( k ) > thr ] > , C ( t ) = | P | (cid:88) ( i , j ) ∈ P M i j ( t ) + W · H − | P | (cid:88) ( i , j ) (cid:60) P (1 − M i j ( t )) , C P = | T P | max t ∈ T P C ( t ); (15)here w =
10 frames = P - a set of pixels belonging ideal pattern, W , H - network dimension, thr - spike threshold, I - indicator function, T P - a set of frames in the tracking range of pattern P . In a sense, this correlation metric can be associated with 1 − d averaged between pattern andbackground, where d is the Hamming distance.
4. Results
Let us show how the neuron-astrocyte network model exhibits memory formation. First, wewill consider a simple single-item memory task illustrating information loading, storage, andretrieval. Next, we will demonstrate how the network can be successfully trained to memorizeand recall several patterns with significant overlaps. Finally, we will analyse model performancemetrics, capacity, and characteristic of pattern remembering on di ff erent parameters. First, we will test the neuron-astrocyte network in the most common experimental paradigmof WM studies - the DMS task. This task requires a single item to be held in memory during abrief delay period. Before specific stimulation, neural network demonstrates irregular, low-ratebackground activity (see activity beginning in Fig 3). At the 500 ms mark, we load an item byapplying transient external input to the corresponding neuronal population for 200 ms (Figs. 2a,3). During training, each astrocyte tracks the activity of neuronal subnetwork associated with it.As soon as the extracellular concentration of glutamate (Figs. 2b,d) and correlated firing in neu-rons achieve a certain level, which satisfies the condition Eq. (9), Ca + concentration in matchingastrocytes elevates (Fig. 2e). In accordance with the experimental data (Bindocci et al., 2017),we tuned the model parameters in such way that the onset of the calcium elevation in astrocytesinduced by synchronous neuronal discharge has a delay of ≤ µ M theastrocyte releases gliotransmitters modulating the synaptic strengths in corresponding locations(Fig. 2). The calcium pulse in astrocyte lasts for several seconds. Its duration determines thelength of the delay interval in the DMS task, during which the item is maintained in the memory.After the training stimulus ends, we test maintenance of the single-item memory by applyingtwo nonmatching items and cue item with t test durations and 250 ms inter-item interval (Figs.2, 3). Because the astrocytic feedback also depends on the activity of neuronal subnetwork, themodel responds di ff erently to the applied items. A short presentation of the cue to the neural net-work evokes the astrocytic-induced increase in the synaptic strength between stimulus-specific11 igure 2: Model of neuron-astrocyte interaction. (a) Spike train and (b) concentration of neurotransmitter, G ( t ), ofstimulus-specific neuron. (c,d) Same as in (a,b) but for an unspecific neuron. (e) Intracellular concentration of Ca + and IP in stimulus-specific astrocyte. Black bars at the top indicate periods when each of the stimuli (training stimulus- sample, nonmatching test items - nonmatch, test cue - match) was presented. In response to presynaptic spike train(a,c), the neurotransmitter, glutamate, G releases (b,d) into extracellular space and the concentration of IP increases inastrocyte (e, blue line) inducing the elevation of intracellular Ca + (e, red line). neurons and results in a local spatial synchronization in the whole stimulus-specific neuronalpopulation (Fig. 2a with comparison to Fig. 2c). Similar to experimental data (Miller et al.,1996), delay activity in our model is sample-selective. We observe that the pattern-specific firingrate in the neuronal network increases and is equal to 270 Hz in comparison with response tonon-specific stimulus (80 Hz) (Fig. 3,b). Such a high frequency is determined by the choiceof fast-spiking neuron model (Izhikevich, 2003). The firing rates in simulations with regularspiking neuron model (Izhikevich, 2003) are almost 10 times lower: 30 Hz for stimulus-specificand 4.5 Hz for non-specific stimulus. The elevation of the frequency in the stimulus-specificneuronal population can continue after the end of the cue, which is determined by duration of theastrocyte-induced enhancement of the synaptic weight.For a visual representation of memory formation, we follow space-time distribution of sample-selective delay activity. Figure (4) illustrates the spatial distribution of activity in neuronal andastrocytic layers at the di ff erent moments of training and cued recall for the same single-itemmemory task as presented in the Figure (3). Training induces the emergence of synchronizedcalcium activity of spatially clustered astrocytes (Fig. 4c). Note that locally synchronised astro-cytes have been found in the neocortex and hippocampus in situ and in vivo (Sasaki et al., 2011;Takata and Hirase, 2008). Such calcium activity correlated in time and space can lead to thespatial-temporal synchronization in the neuronal network (Araque et al., 2014). This mechanismof neuron-astrocyte network interaction underlies the sample-selectivity and pattern retrieval inthe model. Note that 20 % noisy cue item (Fig. 4d) can be identified and cleared from noiseby the neuronal network due to the astrocyte-induced feedback (Fig. 4e). In other words, thespiking neuronal network accompanied by astrocytes can filter a cue pattern distorted by noise.Video of the single-item memory encoding and cued recall in the neuron-astrocyte network canbe found in the supplementary material. 12 igure 3: Delayed matching to sample WM task. (a) One trial of the task simulated in the network. Spike rasterof neuronal network showing sample-selective delay activity. Neurons belonging to stimulus-specific population areindicated by red color. Black bars indicate periods when each of the stimuli was presented. (b, c) The averaged firingrate of the stimulus-specific and unspecific neurons over time, respectively. (20 ms bins)Figure 4: Snapshots of the training (a,b,c) and testing (d,e) neuron-astrocyte network in the single-item working memorytask. (a) Training sample. (b) Response of the neuronal network to the sample. The values of the membrane potentialsare shown. (c) Intracellular Ca + concentrations in astrocytic layer. (d) Testing item with 20% salt and pepper noise. (e)Cued recall in the neuronal network. The firing rate averaged on the test time interval for each neuron is shown. igure 5: Neuron-astrocyte network simulation with four loaded memory items. (a,c,e) Spike train and (b,d,f) concen-tration of neurotransmitter, G ( t ), of three neurons belonging to di ff erent stimulus-specific populations. (a,b) Stimulus-specific neuron to sample 0. (c,d) Stimulus-specific neuron to sample 2. (e,f) Neuron unspecific to all samples. Blackbars at the top indicate periods when each of the stimuli (training stimulus - sample, nonmatching test items - nonmatch,test cue - match) was presented. Next, we consider the multi-item WM formation. In this case, we loaded four items, imagesof numerals 0,1,2,3 in the following way. The images were loaded consequently by applyingexternal inputs of t stim durations with 100 ms inter-item intervals (see Figs. 5, 6a). Due tothe coincidence of di ff erent stimulus-specific neuronal populations in space, the spatial calciumpatterns in astrocytic layers for di ff erent items overlap significantly (Fig. 7h). After a 700 mstraining stimulus was applied, we tested the maintenance of the memory by applying matchingand nonmatching items of t test durations and 250 ms inter-item intervals (Figs. 5, 6a). Theimages were distorted by 20% noise. The astrocyte-mediated feedback modulating coherentneuronal activity provided the selectivity of the model response. The system remembered thecorrect image. Thus, we observed that all items were successfully filtered only in the cued recall.Needless to say that firing rate increases significantly in the cued recall due to the selectiveincrease of synaptic strengths (Figs. 6b,c). To evaluate the performance of the neuron-astrocyteWM, we used as metric the correlation between recalled item and the ideal item during the multi-item WM task (see section Memory performance metrics) (Fig. 6d). It is important to note, thatduring multi-item remembering spurious correlations never dominate in that sense accuracy ofour system is always equal to 100%. There was an increase in correlation with the target imageand no attraction to the wrong image or chimeras. Maximal correlation reached 95% in trainingand 93% in testing sets on average for 4 samples. Video of the multi-item WM in the neuron-astrocyte network can be found in the supplementary material 2.To characterise the quality of memory formation in the model, we examined the dependenciesof correlation of retrieval pattern in cued recall on variable parameters of input patterns, astro-cytic, synaptic, and network structure (Figs. 8,9). First, we investigated its dependence on noiseparameters. The dependence of correlation of recalled pattern on the noise level in training andtest experiments is shown in Figure (8a). Specifically, the correlation di ff erence between therecalled pattern and noisy input is presented. In other words, the model can improve test images14 igure 6: Multi-item WM in neuron-astrocyte network. (a) Spike raster of neuronal network with 4 training patterns.Neurons are coloured according to their pattern selectivity. Pattern overlapping in neuronal populations is 35 .
2% onaverage for 4 patterns. Black bars indicate periods when each of the stimuli was presented. (b, c) The averaged firingrate of the stimulus-specific and unspecific neurons over time, respectively. (20 ms bins) (d) Correlation of filtered items.The di ff erent colours correspond to the correlations with di ff erent ideal samples. The dotted line shows the correlationof the testing item (for 20% noise level in test). igure 7: Snapshots of the training (a, b, c, f, g, h) and testing (d, e, i, j) neuron-astrocyte network in the multi-itemworking memory task. We used the following training set consisting of 4 samples: 0,1,2,3. (a,f) The example of first andlast training samples, respectively. (b,g) The response of the neuronal network to samples. The values of the membranepotentials are shown. (c,h) The intracellular Ca + concentrations in astrocytic layer. (d,i) The testing items with 20% ofsalt and pepper noise. (e,j) The cued recalls in the neuronal network. The firing rate averaged on the test time intervalfor each neuron is shown. depending on noise in training and testing. Training the network with samples with a low noiselevel (up to 25%) provides high correlation. The elevation of the noise level in training sampleinduces a random activity pattern in astrocytic network, which in turn leads to noisy recall.The morpho-functional structure of connections between neurons and astrocytes can a ff ectpattern retrieval in the model (Figs. 8b,c). The key parameters determining this structure are thefraction of synchronously spiking neurons of neuronal ensemble corresponding to the astrocyterequired for the emergence of the calcium elevation in the astrocyte, F act , and the fraction ofsynchronously spiking neurons of neuronal ensemble corresponding to the astrocyte required forthe emergence of astrocyte-induced enhancement of synaptic transmission, F astro . Here we donot account for various spatiotemporal properties of gliotransmitter release and astrocyte Ca + signals evoked by di ff erent levels of neuronal activity (Araque et al., 2014). We assume that si-multaneous activation of synapses induce multiple Ca + events at di ff erent processes of astrocyte,which are spatially and temporally integrated and result in the generation of a global, long last-ing Ca + elevation (Bindocci et al., 2017) that can a ff ect synaptic transmission in the territory ofindividual astrocytes. For this purpose, parameters F act and F astro estimate the correlation levelof synapses activity in the model. The optimal range of F astro for correlation of recalled patternis [40-60]% (Fig. 8b). Smaller values of the parameter, F astro , lead to the e ff ect of astrocyte-induced synchronization initiated even by non-stimulus specific uncorrelated noise activity in asmall ensemble of neurons. On the contrary, the use of larger F astro values implies that a highlycorrelated activity of almost all neurons located in the territory of a given astrocyte is requiredfor existence of astrocytic modulation of synapses. Hence, the neuron-astrocytic network can notperform a correct recall of noisy cue. Another point is that the Figure 8b was obtained for train-ing set with a low 5% noise level and did not reveal the dependence of the correlation of recalledpattern on the parameter F act . We studied the influence of the parameter F act on the correlationin the simulations with di ff erent noise in training samples (Fig. 8c). For lower noise level intraining samples, the network memorises items regardless of the value of the parameter, F act .16ncreasing the level of noise in training samples for small values of the parameter, F act , leads tothe Ca + elevation in randomly distributed astrocytes first, and then in the whole astrocytic layer.Nevertheless, such nonstimulus-specific astrocytic activation can result in high correlation of re-called pattern because of the moderate noise level in the cue and optimally chosen value of theparameter F astro . On the contrary, for larger value of the F act >
85% Ca + signal in astrocyte canbe evoked only by relatively ”clean” sample with a small percentage of noise < F act = −
85% was optimal for performing the WM tasks by neuron-astrocytenetwork. In this range astrocyte activations were stimulus-specific and the astrocyte layer couldmemorise training samples with a low noise level. Note that recent experiments in the investi-gation of Ca + activity in astrocytes with precise spatial-temporal resolutions (Bindocci et al.,2017) revealed that global calcium event originated from the multiple foci on the most structuralpart of astrocyte periphery depends on the synchronous neuronal discharges.Next, we studied the influence of synaptic connectivity architecture in the neural network,specifically the number, weight, and distribution of synaptic connections, on the correlation inthe multi-item WM task (Fig.9). The minimal number of synaptic connections, N out , requiredfor existence of cued recall is 20 (Fig.9a). A smaller number of connections is not enough toactivate all the neurons from stimulus-specific population. Simultaneous increase of weightsand number of connections induces the generation of large synaptic currents resulting in self-sustained overactivation of neuron-astrocyte network. Therefore there exists the optimal rangeof synaptic weight values to ensure high correlation. We found that for our model this range is η ∈ [0 . − . λ from Eq.(4), the lowerthe probability of long-distance connections. The highest correlation was observed for localconnections, λ <
7, due to the fact that short-range connections do not lead to blurring of thepattern retrieval boundaries in the neural network. Figure 9b also determined the optimal rangefor the number of synaptic connections: N out ∈ [25 , -evoked Ca + -induced Ca + release from the astrocyte endoplas-mic reticulum stores, which is described by the biophysical model (Li and Rinzel, 1994) used inthis study. Fragmentary experimental data on duration of the gliotransmitter-induced modulationof synaptic transmission shows that the short-lived version of this modulation lasts from fractionsof a second to few minutes, while long-term plasticity can last for tens of minutes (for review see(Pitt`a et al., 2016)).To characterise memory capacity, we subjected it to longer trains of samples. Samples wereapplied to neuronal ensembles with average 35 .
2% overlapping in population. To check the mem-orisation, we presented cue items in the reverse order compared to learning mode (e.g. learning:0 , , , ..., ,
8; test: 8 , , ..., , , τ astro , onthe capacity. The number of items stored in the system memory is maximum for parameter val-17es in the range: τ astro ∈ [60 , τ astro excitation does not have timeto spread to the stimulus-specific neural population; for long astrocytic modulation the di ff erentitems in cued recalls interfered with each other.In the case of non-overlapping neuronal ensembles, the WM capacity is unequivocally de-termined by the duration of the astrocytic calcium signal and can be obtained analytically. Weestimated the Ca + signal duration in τ Ca = . K sam-ples of a fixed order: 1 , , ..., K . During the test, we look at a permutation p consisting of K patterns. For the permutation, we estimate the number of correctly recalled patterns, K p . Patternis considered correctly recalled if no more than τ astro has passed since its presentation. The aver-age capacity, C , for this case is defined as the average number of correctly recalled patterns overall possible permutations p . According to this description the average capacity can be calculatedby the following equation: t itrain = i · τ + ( i − τ , t j , itest = t itrain + τ shi f t + j · τ + ( j − τ , C = K ! (cid:88) p K (cid:88) (cid:104) t p i , itest − t itrain < τ Ca (cid:105) = K ! K (cid:88) (cid:88) p (cid:104) t p i , itest − t itrain < τ Ca (cid:105) = ( K − K ! K (cid:88) K (cid:88) j = (cid:104) t j , itest − t itrain < τ Ca (cid:105) , (16)where τ - train sample duration, τ - duration between train samples, τ - test sample duration, τ - duration between test samples, τ shi f t - delay between train and test, τ Ca - calcium eventduration, t itrain - time of i-th train sample finishes, t j , itest . Figure 11 shows the capacity as a functionof the sample number, K . After it reaches a maximum of 6.6 for 8 samples, the capacity beginsto decrease monotonically, while the number of samples increases.
5. Discussion
We proposed a biologycally motivated spiking neuron network model accompanied by astro-cytes that demonstrates the working memory formation. The model acts at multiple timescales:at a millisecond scale of firing neurons and the second scale of calcium dynamics in astrocyte.Neuronal network consists of randomly sparsely connected excitatory spiking neurons with non-plastic synapses. Astrocyte induced activity-dependent short-term synaptic plasticity results inlocal spatial synchronization in neuronal ensembles. WM realised by such astrocytic modulationis characterised by one-shot learning and is maintained during seconds. The astrocyte influenceon the synaptic connections during the elevation of calcium concentration implements Hebbian-like synaptic plasticity di ff erentiating between specific and non-specific activations. Note that theproposed model is crucially di ff erent from the attractor-based network memory models (Hop-field, 1982; DJ, 1995; Wang, 2001; Wimmer et al., 2014) and works similarly to WM modelsbased on synaptic plasticity (Mongillo et al., 2008; Manohar et al., 2019; Lundqvist et al., 2011;Mi et al., 2017). In particular, in its functionality, the model is quite close to short-term associa-tive (Hebbian) synaptic facilitation (Fiebig and Lansner, 2016; Sandberg et al., 2003).18 igure 8: Correlation between recalled pattern (see section 3.5) and ideal item in a multi-item WM task performed by theneuron-astrocyte network. The correlation averaged over 4 patterns is shown. (a) Noise-resistance ability of the model.Dependence of correlation on noise level in training and test. The correlation di ff erence between cued recall pattern andnoisy input is shown. (b),(c) The influence of the neuron-astrocytic interaction structure. (b) Dependence of correlationon the number of spiking neurons required for the calcium elevation in the astrocyte, F act , and on the number of spikingneurons required for the emergence of astrocyte-induced enhancement synaptic transmission, F astro . (c) Dependence ofcorrelation on noise level in training samples and on the parameter, F act . F astro = .
5. For (b,c) noise level in cue is20%. igure 9: Influence of synaptic connectivities architecture in the neural network on the correlation of recalled patternin multi-item WM task performed by the neuron-astrocyte network. The correlation averaged over 4 patterns is shown.(a) Dependence of correlation on the number of output synaptic connections for each neuron, N out , and synaptic weight, η . (b) Dependence of correlation on the number of output synaptic connections for each neuron, N out , and synapticconnection distribution parameter, λ Eq.(4).Figure 10: Capacity of the multi-item WM in neuron-astrocyte network. Capacity as a function of the sample number intraining. The number of images with correlation of recalled pattern higher than 90% is shown. igure 11: Capacity of the multi-item WM in neuron-astrocyte network in the case of non-overlapping neuronal popula-tions obtained analytically. Capacity as a function of the sample number in training sequence. Learning Storage Retrieval Ca pattern astrocytes neurons synchronizedactivity
Input Ca pattern astrocytes neurons synchronizedactivity Ca pattern astrocytes neurons
OutputCue
Figure 12: Concept of WM operation in spiking neuron network model accompanied by astrocytes. ffi cient degree of stability, which ensures mem-ory retention despite the presence of significant overlaps in the stimulus-specific subnetworks.Needless, to say that astrocyte-induced modulation of synaptic transmission proposed in thisstudy as a mechanism for WM organisation does not exclude but rather complements othersynaptic and neural plasticity mechanisms (fast Hebbian synaptic plasticity / short-term synap-tic plasticity, facilitation, augmentation, dendritic voltage bistability, etc.) and may well act inparallel to them.On the one hand, there has been much experimental evidence that astrocytes contribute tosynaptic plasticity, coordination of neural network oscillatory activity, and memory function(Santello et al., 2019). It was shown recently that astrocytic impact is circuit-specific (Martinet al., 2015) and stimulus-specific (Mariotti et al., 2018). Improved Ca + imaging approacheshave identified a spatiotemporal diversity of astrocytic signals that may underlie the capacity ofastrocytes to encode and process di ff erent patterns of activation (Bindocci et al., 2017; Stobartet al., 2018). Besides, the temporal scale of the astrocytic calcium dynamics and dynamics ofthe neuron-astrocyte bidirectional communication including the e ff ects of astrocytic influence onsynaptic plasticity fit very well in timing required in WM processes.On the other hand, the ongoing intense debate about principles of WM organisation challengesthe canonical theory of persistent delay activity in network attractors with recurrent excitation(Bouchacourt and Buschman, 2019) and o ff er alternative models incorporating di ff erent bio-physical network mechanisms of WM (Lundqvist et al., 2018; Barak and Tsodyks, 2014). Theprincipal reasons for such a debate is the reexamination of experimental data, which shows alarge heterogeneity in the delay neuronal activity during WM tasks (Stokes et al., 2013). A non-classical WM model includes the short-term synaptic plasticity (Mongillo et al., 2008; Hanseland Mato, 2013), the balance of inhibition and excitation (Boerlin et al., 2013), the NMDAcurrents a ff ecting on the neuronal excitability (Durstewitz, 2009), and other parameters. Thesemodels, however, have a number of shortcomings: inability to describe encoding of novel asso-22iations in synaptic facilitation-based model; unclear mechanisms for achieving precise tuningof recurrent excitation and inhibition; the time constant of the NMDA receptor is appropriateto maintain memories for 1–5 s, but not longer.The investigation of the synaptic mechanismsunderlying WM is an ongoing process. Therefore, incorporation of astrocytes as spatiotemporalintegrator and modulator of synaptic transmission in neural network models may help advancethe theoretical framework of WM encoding and maintenance mechanisms.Future research direction in the framework of the proposed WM model in the neuron-astrocytenetwork will be focused on the interplay of excitation and inhibition that can stabilise WM[100]; the e ff ects of synaptic plasticity (Boerlin et al., 2013); the e ff ects of synaptic plasticity(Hansel and Mato, 2013; Mongillo et al., 2008) namely associative short-term potentiation (afast-expressing form of Hebbian synaptic plasticity) that can provide an encoding of novel asso-ciations (Fiebig and Lansner, 2016); and on the structure of the cortical microcircuit reflectingcolumnar organisation of the neocortex.This research was supported by the Ministry of Science and Higher Education of the RussianFederation (project No. 075-15-2020-808, No. 0729-2020-0061). SG thanks the RFBR (grantNo. 20-32-70081). References
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