Computational simulation of dentate gyrus granule

Neurocomputing 175 (2016) 300–309

Contents lists available at ScienceDirect

Neurocomputing journal homepage: www.elsevier.com/locate/neucom

Computational simulation of dentate gyrus granule cell—The role of metaplasticity Nicholas Hananeia n, Lubica Benuskova Department of Computer Science & Brain Health Research Centre, University of Otago, Owheo Building, 133 Union Street East, Dunedin 9016, New Zealand

art ic l e i nf o

a b s t r a c t

Article history: Received 10 October 2014 Received in revised form 6 October 2015 Accepted 19 October 2015 Communicated by J. Torres Available online 30 October 2015

After several decades of study, the dynamics of synaptic plasticity in neurons still remains somewhat a mystery. By conducting a series of simulations on a simulated version of an in-vivo experiment on the rat dentate gyrus granule cell, using the Izhikevich spiking neuron model, we compare and contrast several potential synaptic plasticity rules' applicability to the same experiment. Our simulations reveal that spike timing dependent plasticity (STDP), a more recent theory of synaptic plasticity, is insufficient to replicate the heterosynaptic LTD shown in the experiment without including aspects of the significantly older Bienenstock–Cooper–Munro (BCM) theory. The STDP rule modified by including the history of postsynaptic spiking seems most likely to be an accurate candidate for reproducing the heterosynaptic plasticity dynamics. & 2015 Elsevier B.V. All rights reserved.

Keywords: Synaptic plasticity Metaplasticity STDP BCM

1. Introduction Current understanding of the mechanisms of learning and longterm memory storage in the brain implies a key role for changes in synaptic weights induced by coincident pre- and postsynaptic activity [2]. In many regions of the brain, long-term potentiation (LTP), a prolonged increase in the synaptic efficacy of excitatory synapses, is produced by high-frequency stimulation (HFS) of presynaptic axons [3]. LTP in the rat can be either short- (1–3 h), intermediate- or long-lasting (4 24 h), depending on the HFS protocol administered to the presynaptic inputs [4]. On the other hand, lowfrequency stimulation (LFS) can yield long-term depression of synaptic weights [5], but often yields no change [6], and recently even a LFS induced LTP has been documented [7]. In order to reconcile numerous counter-intuitive results of frequency-dependent synaptic plasticity, Bienenstock, Cooper and Munro (BCM) proposed an influential theory of synaptic plasticity to explain plasticity in the developing visual cortex [8], which was later shown to hold in the adult somatosensory cortex too [10]. The crucial notion in the BCM theory is the existence of a so-called sliding LTD/LTP threshold. The LTD/LTP threshold corresponds to a value of the frequency of presynaptic stimulation, below which the stimulation induces LTD and above which the stimulation induces LTP. In addition, the position of the LTD/LTP threshold is not fixed but instead moves (slides) in proportion to the average postsynaptic activity. When the neuron is n

Corresponding author. E-mail addresses: [email protected] (N. Hananeia), [email protected] (L. Benuskova). http://dx.doi.org/10.1016/j.neucom.2015.10.063 0925-2312/& 2015 Elsevier B.V. All rights reserved.

more active on average, the LTD/LTP threshold slides to higher values and it is more difficult to get LTP and easier to get LTD. The opposite is true when the average postsynaptic activity is low. In the BCM models, neurons received not only the patterned stimulation but also the noise corresponding to an ongoing spontaneous activity in the neural circuits [30,10]. In fact, this noise was “responsible” for weakening synapses that did not receive the patterned activity [9]. Later, many experimental studies demonstrated that prior history of pre- and postsynaptic neural activity controls the subsequent induction of LTP and LTD. This phenomenon is known as metaplasticity [11]. Metaplasticity thus refers to the prior history of pre- and/ or postsynaptic neural activity controlling the occurrence and magnitude of subsequent induction of synaptic plasticity. In addition, there are numerous experimental studies showing that both LTP and LTD depend not only on the frequency of the presynaptic stimulation but also on the precise timing of pre- and postsynaptic spikes. This property is called spike-timing-dependent plasticity or STDP for short [13]. Presynaptic spikes that precede postsynaptic spikes within a certain time window produce LTP, whereas presynaptic spikes that follow postsynaptic spikes within a certain time window produce LTD of synapses. Experimentally observed positive and negative changes in synaptic weight w are best fitted with exponential relationships, i.e.

Δw þ ¼ A þ expð Δt=τ þ Þ if Δt 4 0

ð1Þ

Δw ¼ A expð Δt=τ Þ if Δt o 0

ð2Þ

where Δt ¼ t post t pre is the time difference between the post- and presynaptic spikes. Amplitudes A þ , A and decay time constants

N. Hananeia, L. Benuskova / Neurocomputing 175 (2016) 300–309

τ þ , τ for synaptic potentiation and depression, respectively, are different for different species, brain areas, and other conditions of experiment. We will refer to Eqs. (1) and (2) as the STDP rule. The final synaptic change can be additive (i.e. positive and negative changes add over time) or multiplicative (i.e. positive and negative changes multiply over time). There are also different options as to how many and which pre- and postsynaptic spikes contribute to the final change [23]. One of the first attempts to connect the frequency- and timingdependent synaptic plasticity in a single phenomenological model was made by Sjöström et al. [18]. Later, more variations of the STDP rule accounting for the frequency-dependent synaptic plasticity were proposed (see, e.g. [14]). One of such modifications of the original rule (Eqs. (1) and (2)) is the STDP rule with metaplasticity introduced by Benuskova and Abraham in 2007 [15]. The authors have brought STDP, frequency-dependent plasticity and metaplasticity into a unified theoretical framework and provided putative explanation of heterosynaptic plasticity phenomena in the hippocampal dentate gyrus of freely moving rats as reported in [16]. Heterosynaptic plasticity means that high-frequency stimulation (HFS) of one set of synapses leads to synaptic plasticity not only of the stimulated synapses but also in a neighbouring set of synapses that were not subject to HFS [16]. This heterosynaptic plasticity is still a puzzling matter. The reasoning behind their modification of the STDP rule was based on the paper of Izhikevich and Desai [17], in which the authors showed mathematically for uncorrelated and weakly correlated Poisson spike trains that the STDP, Eqs. (1) and (2), actually lead to the emergence of a fixed LTD/LTP frequency threshold, but only when we consider the nearest-neighbour spike interactions. To include the sliding property of the LTD/LTP threshold as a function of previous postsynaptic activity, Benuskova and Abraham suggested that amplitudes, A þ , A , of LTP and LTD, respectively, are not constant but instead depend metaplastically on the average postsynaptic spiking activity [15]. They used this new STDP with metaplasticity to explain the frequency-dependent homosynaptic LTP and heterosynaptic LTD demonstrated in the dentate gyrus of the hippocampus of live rats [16]. Thus, they assumed that STDP can underlie also the frequency-dependent synaptic plasticity. This assumption is corroborated by the experimental study of Lin et al. [19], who published experimental results on STDP in granule cells in hippocampal slices. Lin et al. used pairs of presynaptic stimuli and postsynaptic antidromic spikes delivered to granule cells in different orders (pre–post and post–pre) and with different delays to successfully induce STDP. Stimulated synapses exhibited STDP with two windows, one for LTP for the pre–post sequence and the other one for LTD for the post–pre sequence. In addition, Lin et al. [19] showed the interaction of STDP and frequency-dependent plasticity at one synaptic path, thus suggesting that they may actually share the same biological mechanisms. In order to investigate the role of metaplasticity in explaining the heterosynaptic plasticity phenomenon, we have implemented and simulated the spiking model of the granule cell in the hippocampal dentate gyrus. We aim to reproduce the homosynaptic long-term potentiation of the tetanised input and heterosynaptic long-term depression of the untetanised input, as observed in real experiments after applying HFS to the MPP input [16] with a selection of STDP-inspired models of synaptic plasticity. We use the Benuskova & Abraham rule [15] as a baseline, to which the other rules are compared. The other rules we have implemented are the original, unmodified STDP rule (hereafter referred to as conventional STDP), the Froemke et al. suppression model [20], Pfister & Gerstner's triplet STDP model [21], and the Clopath et al. voltage-dependent model [22]. The latter model allows for metaplastic modification of the LTD amplitude only, and we will test Pfister & Gerstner's model alongside a modified version of itself

301

allowing for metaplastic modification of both STDP amplitudes. Our results show that all of the STDP-like plasticity rules conformed closest to experiment feature at least some level of BCMlike metaplasticity.

2. Methods 2.1. Spiking neuron model In this study, we simulated a spiking model of excitatory granule cells (GCs) in the dentate gyrus. The dentate gyrus is the input part of the hippocampus [25]. Granule cells in the dentate gyrus receive excitatory input from outside of the hippocampus via the lateral and medial perforant paths. The medial and lateral perforant paths (MPP and LPP respectively) are two separate inputs terminating on separate but adjacent dendritic zones of the hippocampal dentate granule cells. The MPP and LPP are the major excitatory inputs to the GC as their synapses occur at more than 80% of the GC dendritic tree [24]. In turn, granule cells project their axons to the pyramidal cells in the CA3 area of the hippocampus, thus relaying the spatial and associational information for further processing in hippocampal circuitry. We used a simple model of a representative dentate granule cell (GC), in which we ignored the effects of local inhibitory as well as local and contralateral excitatory neurons. Thus, the model neuron has only two inputs representing the medial and lateral perforant paths. In the real GC, the MPP synapses are closer to the soma as they are localised on the medial part of the dendritic tree while LPP synapses are localised on the most distal part [25]. As a result, there is a delay in propagation of PSPs and backpropagtion of action potentials (bAPs) in the dendritic tree. However, GCs are small compact cells in comparison with, for instance, pyramidal cells. Modelling studies using the multicompartmental model of the GC dendritic tree show that the delay between bAPs reaching MPP and LPP synapses is about 1–2 ms (see e.g., Fig. 3B in [12]). Thus, we have neglected this small delay in the model of the spiking neuron and in implementations of all STDP rules below. For the neuron model we employed the simple model of a spiking neuron introduced by Izhikevich [26]. The neuron model is described by two dimensionless variables v(t) and u(t) obeying these two ordinary differential equations: dv ¼ 0:04v2 þ 5v þ 140 u þ I dt

ð3Þ

du ¼ aðbv uÞ dt

ð4Þ

Variable v corresponds to membrane voltage and u is the socalled recovery variable. After the value of variable v reaches spike apex (e.g., AP¼ 55 mV), the membrane voltage and the recovery variable are reset according to the formula: ( v’c If v Z AP then ð5Þ u’u þ d Different firing characteristics of neurons (i.e. regular spiking, chattering, and bursting) are achieved with different values of dimensionless parameters a, b, c and d. We have employed the parameter values corresponding to a regularly spiking excitatory cell, because this is appropriate for granule cells, i.e. a ¼ 0:02; b ¼ 0:2; c ¼ 69 mV; d ¼ 2. Synaptic inputs are delivered via variable I, which is determined in the same way for spontaneous and evoked input activity, i.e. I ¼ sMPP wMPP N MPP þ sLPP wLPP NLPP

ð6Þ

We update the neuron model every 1 ms. That is, we numerically calculate/evaluate Eqs. (3)–(6) for every millisecond of real

302


time. When there is an input spike, evoked or spontaneous, at the corresponding synapse, be it LPP or MPP, the function sðtÞ ¼ 1 for that time moment and pathway, otherwise sðtÞ ¼ 0. Parameter NMPP (NLPP) is the number of MPP (LPP) fibres, respectively, engaged by presynaptic stimulation. Variable wMPP (wLPP) is the weight of the MPP (LPP) synapse, respectively. In our simulations, during test stimulation, N MPP ¼ N LPP ¼ 150, and during HFS and spontaneous activity, N MPP ¼ N LPP ¼ 250. These values reflect the same ratios of stimulating intensities delivered to the MPP and LPP in the real experiment [16]. The initial values of synaptic weights were chosen to be wMPP ¼ wLPP ¼ 0:03, so when the two input pathways MPP and LPP were stimulated, either simultaneously or in close temporal succession, a postsynaptic spike followed. 2.2. Simulation of presynaptic activity Because the simulated experiments were done in awake animals, our model granule cell was subject to the ongoing spontaneous input activity that is present in biological neurons in vivo. All the spontaneous input spiking were generated randomly as Poisson spike trains. The Poisson process is the simplest model of neuronal firing and is often found in recordings of the spontaneous activity of biological neurons in vivo [27]. Simulated ongoing spontaneous input spiking had two components. The first component was the correlated spontaneous spikes in the MPP and LPP. By correlated we mean that the spikes occurred at the same time in both inputs. Superimposed on this correlated spiking was low-frequency (0.1 Hz) uncorrelated random activity occurring at different times in both inputs. The average frequency of all spontaneous input spiking at both inputs was 8 Hz to simulate the theta frequency observed experimentally at the axons from the entorhinal cortex in live rats [28]. The simulated spontaneous activities, as described above, have led to a postsynaptic spontaneous spiking activity of the simulated granule cell of around 1 Hz (for all values of parameters as used below), which is in accordance with the data for granule cells [29]. Superimposed upon the spontaneous input were the low-frequency test spikes delivered alternately to MPP and LPP at 10-s intervals 30 min before and 90 min after HFS of the MPP [16]. The reason for the test stimulation is that in real experiments, the researchers cannot observe the changes in synaptic weights directly, but need to use a low intensity stimulation to evoke a test response from the neuron (i.e. excitatory postsynaptic potential), from which they derive the changes in synaptic weights. Superimposed on spontaneous input spiking was also the highfrequency stimulation (HFS) of MPP input. In the real experiment as well as in our computer model, MPP HFS consisted of 50 trains of ten pulses at 400 Hz, delivered after 30 min of test stimulation. HFS was delivered in bursts of 5 trains at 1-s intervals, with 60 s between bursts. The time required to complete the HFS was about 10 min of real and simulated time. During the trains of HFS at the MPP, there was an ongoing 8 Hz-spontaneous input activity of LPP input only. Between the HFS trains and bursts, however, both inputs received uncorrelated spikes generated with an average frequency of 8 Hz. All types of input-output spiking activity, be it spontaneous, test or HFS, were accompanied by STDP as described below. 2.3. STDP with metaplasticity – Benuskova & Abraham rule For the purpose of a baseline with which to compare our selection of STDP rules, we use the STDP rule endowed with metaplasticity as in [15]. For the STDP rule we used a formula presumably holding for hippocampal excitatory neurons, in which presynaptic spikes that precede (follow) postsynaptic spikes within a certain time window produce long-term strengthening (weakening) of synapses [19]. Thus, the positive and negative synaptic changes, Δw þ and Δw , are calculated according to Eqs. (1) and (2). For each presynaptic

spike only two postsynaptic spikes are considered: the one that occurs before and the one that occurs after the presynaptic spike. The total weight is calculated as wðt þ 1Þ ¼ wðtÞð1 þ Δw þ Δw Þ [17]. Benuskova & Abraham [15], proposed that the STDP amplitudes of positive and negative synaptic changes, A þ and A , are not fixed, but instead they dynamically change as a function of the average of the postsynaptic spiking activity over some recent past 〈c〉 such that at each time instant t: A þ ð0Þ A þ ðtÞ ¼ cðtÞ

ð7Þ

A ðtÞ ¼ A ð0Þ cðtÞ

ð8Þ

Positive constants A þ ð0Þ and A ð0Þ are initial amplitude values for synaptic potentiation and depression, respectively. Eqs. (7) and (8) simply mean the amplitude for LTP gets smaller and the LTD amplitude gets larger when the average postsynaptic activity is high. The opposite is true for a low average postsynaptic activity. Then, it is easier to potentiate the synapses than to weaken them due to an expanded amplitude for LTP and shrunken amplitude for LTD. The new values of A þ ðtÞ and A ðtÞ are updated at each iteration based on the actual value of the recent average activity 〈c〉. Average postsynaptic activity 〈c〉 is calculated as an integral: Z c0 t 0 cðtÞ ¼ cðt 0 Þ expð ðt t 0 Þ=τÞ dt ð9Þ

τ

1

with c0 ðtÞ ¼ 1 or 0 if the postsynaptic spike is present or absent at time t 0 , respectively, τ is the integration period, and c0 is the scaling constant. We used the value of τ ¼ 60,000 ms ¼1 min for all simulations. The integral can be replaced with a discrete sum, but we actually numerically calculated the above integral in our code. We did not have hard upper or lower bounds for the weights, nor did we do renormalisation, instead we employed a soft stability control in the form of this condition: if 〈c〉 o 0:01 OR 〈c〉4 100, then A þ ðtÞ ¼ A þ ðt 1Þ, and A ðtÞ ¼ A ðt 1Þ. The rationale for using the spike count for Eq. (9) comes from experiments of Abraham et al. (2001), in which antidromic spikes (with NMDA receptors blocked) were sufficient to increase the threshold for subsequent LTP induction by HFS. In this calculation of the average of the past spike count of the postsynaptic cell, the most recent spikes have the largest weight and this influence decays exponentially into the past. This relationship was inspired by the calculation of the dynamic position of the LTD/LTP threshold in the plasticity model of the visual [30] and somatosensory cortices [10]. 2.4. Froemke et al. rule Froemke et al. proposed [20] a “suppression model” for STDP that implements a sort of metaplasticity for layer 2/3 pyramidal cells in vitro. In the latter paper, they compared the basic “historyindependent” STDP model (i.e. one without any form of metaplasticity) to their suppression model. The suppression model is implemented as a dynamic scaling factor to the original STDP rule, which Froemke et al. call FðΔtÞ. The weights are changed as such: ( A þ expð Δt =τ þ Þ if Δt 40 FðΔtÞ ¼ ð10Þ A expð Δt =τ Þ if Δt o0

Δw ¼ ϵpre ϵpost FðΔtÞ

ð11Þ

For both pre- and postsynaptic spikes it holds that

ϵk ¼ 1 expð ðt k t k 1 Þ=τs Þ

ð12Þ

where τs is a decay constant for the suppression effect, tk and t k 1 are the timings of spikes; presynaptic if it is for ϵpre and postsynaptic if it is for ϵpost. As can be seen, the scaling factors for the presynaptic and postsynaptic neurons are calculated separately


based on when the neurons spiked, and then multiplied with the basic STDP weight change. The bigger the frequency of spikes, the smaller the suppression factors will get, and the smaller the weight changes become. The duration of this effect is adjusted with τs. Although this effect is metaplastic in nature, i.e. with high frequencies the growth of weights ceases, and with low frequencies, the weight increase is encouraged, it is by no means BCM-like. No hard-coded limits on weight are necessary here, since the suppression effect limits weight increases. Eqs. (10)–(12) are the so-called “original suppression model”. Froemke et al. proposed also a “revised suppression model” to account for the data from bursting (cortical) cells [20]. Since neither granule cells nor their afferents are bursting, we did not implement this revised synaptic plasticity model.

events – instead it makes changes to the weights based on the cell's membrane voltage at each time step. In this model, LTP and LTD are described by the differential equations: dw ðtÞ ¼ A ðhuiÞXðtÞðu ðtÞ θ Þ dt dw þ ðtÞ ¼ A þ xðtÞðu θ þ Þðu þ θ Þ dt

wðt þ 1Þ ¼ wðtÞ o1 ðtÞ½A2 þA3 r 2 ðt ϵÞ

if t ¼ t pre

ð13Þ

wðt þ 1Þ ¼ wðtÞ þ r 1 ðtÞ½A2þ þ A3þ o2 ðt ϵÞ

if t ¼ t post

ð14Þ

As can be seen, in addition to the amplitudes A2 and A3, inside these two weight change equations are embedded four functions which are described by the four differential equations: dr 1 ðtÞ r 1 ðtÞ ¼ ; dt τþ

if t ¼ t pre then r 1 -r 1 þ1

dr 2 ðtÞ r 2 ðtÞ ¼ ; dt τx

if t ¼ t pre then r 2 -r 2 þ1

ð16Þ

do1 ðtÞ o1 ðtÞ ¼ ; dt τ

if t ¼ t post then o1 -o1 þ 1

ð17Þ

do2 ðtÞ o2 ðtÞ ¼ ; dt τy

if t ¼ t post then o2 -o2 þ 1

ð18Þ

ð15Þ

All four of these variables behave similarly, and each decays exponentially according to their own decay constant. These variables serve to track the behaviour of the cell, as each of them is also incremented in value by 1 with each spike. This serves to facilitate the triplet interactions. The Pfister & Gerstner rule does not use any form of weight limit. These differential equations are solved numerically at each time step of the simulation, with the additional updates put in place as spikes occur. 2.6. Pfister & Gerstner rule with metaplasticity We also experimented with a modified version of the Pfister & Gerstner rule, endowed with metaplasticity of the same type as in the Benuskova & Abraham rule [15]. As in the Benuskova & Abraham rule, A2þ , A2 , A3þ , and A3 are not constant, but are instead updated at each iteration using the value of 〈c〉, obtained identically as in our Benuskova & Abraham implementation. 2.7. Clopath et al. rule This model [22] is unlike the other models mentioned, in that this model does not consider neuronal spikes as discrete

if w 4wmin if w o wmax

ð19Þ ð20Þ

These equations contain additional variables described by these differential equations:

τ

du ðtÞ ¼ u ðtÞ þu dt

ð21Þ

τþ

du þ ðtÞ ¼ u þ ðtÞ þu dt

ð22Þ

2.5. Pfister & Gerstner rule This rule does not consider pairs of spikes, but triplets, and, as such, is completely different in form to the aforementioned ones. The rule [21] proposes that in addition to the pairs of spikes that have been considered in the previous rules, there is also a lesser effect on the overall weight change coming from a potential third spike in this association. This rule updates the weights according to the equations:

303

τx

dxðtÞ ¼ xðtÞ þ XðtÞ dt

ð23Þ

hui2 hui2ref

ð24Þ

A ðhuiÞ ¼ A ð0Þ

Here the variable u is the membrane voltage, θ and θ þ are free parameters (LTD and LTP thresholds, respectively), A þ and A are the LTP and LTD amplitudes, τ , τ þ and τx are decay constants. A ð0Þ is the initial LTD amplitude, and hui is a moving average of the membrane voltage, scaled with the inverse of a reference value. X(t) is a variable which is set to 1 whenever there is a presynaptic spike, and reset to 0 when there is not one in the preceding time step. The overall change in synaptic weight is calculated as dwðtÞ dw þ ðtÞ dw ðtÞ ¼ þ dt dt dt

ð25Þ

with the hard bounds wmin r w r wmax . It is worth noting that the Clopath rule treats metaplasticity differently for LTP and LTD. As seen in Eq. (24), metaplasticity for LTD is similar to that used in Benuskova & Abraham, whereas the amplitude for LTP is constant.

3. Results 3.1. Benuskova & Abraham rule As was seen in earlier work [15], this rule is capable of reproducing the results in [16]. To obtain these results, we used the parameters A þ ¼ 0:02, A ¼ 0:01, τ þ ¼ 20 ms, τ ¼ 70 ms, c0 ¼ 2000 and τ ¼ 60 s. The average GC firing frequency is 0.87 Hz (S.D. 0.0019 Hz). Results of the heterosynaptic-plasticity experiment [16] are faithfully reproduced by the model (see the weight evolution Fig. 1, left). HFS commences at 30 min and lasts about 10 min. After commencement of HFS we can see the development of longlasting homosynaptic LTP of the tetanised MPP and heterosynaptic LTD of the untetanised LPP.1 In all figures, in which we depict the evolution of weights, we start showing them from time 0, when the simulated test pulses accompany the spontaneous activity. We do not show the previous time interval (E 1 h) when only spontaneous input spiking is present and the system has reached an equilibrium. 1 It is worth noting that for Benuskova & Abraham rule, there are rare cases when the LTP induction protocol causes the opposite of what we would expect to occur – i.e. potentiation of the LPP and depression of the MPP. These “inverted” cases are quite rare (31 in 1000) and thus can be dismissed when they occur.

304


Fig. 1. Benuskova & Abraham rule. Left: evolution of MPP (red) and LPP (green) weights over time. Arrow indicates the onset of HFS at MPP at 30 min. HFS lasts for 10 min. Average of 30 runs 7 S.D.; Right: evolution of a running average of postsynaptic spike count 〈c〉 7 S:D: Note that during HFS there is a drop in average postsynaptic activity, which has important consequences for the outcome of heterosynaptic plasticity simulation (see text for more details). (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

3.2. Conventional STDP To show that STDP on its own as originally described [13] is insufficient for reproducing our experimental data, we implement the rule as written in Eqs. (1) and (2), with constant amplitudes and with a decay term, i.e. Fig. 2. Benuskova & Abraham rule. Postsynaptic voltage from 10 s before and 10 s after the onset of HFS. During the first 5 s, the first burst of HFS consisting of 5 trains of ten 400 Hz pulses is delivered, followed by a 1-min pause. The trace is from a single run.

This homosynaptic LTP and heterosynaptic LTD is only seen under specific conditions, as explored in Benuskova & Abraham (2007): (1) Spontaneous input spiking is correlated between MPP and LPP inputs before and after HFS. In Fig. 2 we can see that during 10 s before HFS only 6 postsynaptic spikes were generated, thus even correlated presynaptic spiking does not always lead to generation of the postsynaptic spike. (2) Spontaneous input spiking is temporarily decorrelated between MPP and LPP inputs during HFS. One train of ten 400 Hz HFS pulses lasts for 22.5 ms. One burst consists of 5 trains delivered at 1-s intervals, then a pause of 1 min follows. In Fig. 2, we can see that during the first 5 s of HFS, only 4 postsynaptic spikes were generated in response to 5 trains of ten pulses of 400 Hz frequency. Then, due to the decorrelation of spontaneous spiking between MPP and LPP between trains and bursts, no postsynaptic spikes occur. Thus, since our integration period for 〈c〉 is 1 min, the result is that on average there is a drop in running average of postsynaptic activity as we can see on the left in Fig. 1 on the right. (3) The temporary drop in 〈c〉 causes A þ to increase (Eq. (7)), thus those few postsynaptic spikes that are evoked by HFS lead to potentiation of the tetanised MPP input because they occur in a pre–post order. Although the drop in 〈c〉 causes A to decrease (Eq. (8)), LPP weight weakens because random presynaptic spiking of the LPP input does not spike the cell. Although presynaptic spikes occur at random, and thus there should be zero overall change, depression prevails, because the LTD window is much longer than the LTP window. Thus, Benuskova & Abraham's metaplastic STDP rule has a nice homeostatic property, similar to the original BCM rule when applied to the experiments with one eye closed and one eye open during the critical period of the visual cortex development. That is, the synapses relaying signals from an open eye potentiated, while synapses relaying only noise from the closed eye weakened [8,30].

wðt þ 1Þ ¼ wðtÞ λwðtÞ

ð26Þ

We used the following parameter values: A þ ¼ 0:02, A ¼ 0:01, τ þ ¼ 20 ms, τ ¼ 70 ms, 10 5 r λ r 10 3 . With no hard bounds, for λ ¼ 10 3 the average cell firing frequency with these parameters in this STDP model is 8.4 Hz (S.D. 0.029 Hz). For other values of λ it was still very high, i.e. around 8 Hz (Fig. 3). As can be clearly seen, STDP with the decay term does not reproduce the experimental results. After the simulation starts, the weights begin oscillating with increasing amplitude of oscillation and they never stabilise. Upon the HFS onset, we can see the transient potentiation of both the MPP and LPP weights lasting only for the duration of HFS. This transient increase in input weights is accompanied by a transient increase of the average postsynaptic spiking activity. After cessation of HFS, the weight continues to diverge until finally coming to stable values, one of them being always very close to zero. No set of parameters that we used was able to reproduce the experimental results. Fig. 4 shows the postsynaptic voltage trace from a single run, from which we cannot determine too much except that it seems there are more postsynaptic spikes after the onset of HFS. If λ 4 10 3 both weights monotonically decreased until they reached a stable value of 0.004. HFS had no effect. Another variant of the conventional STDP rule we implemented was one without the decay term and where only the hard bounds on weights were imposed. With this rule, the weights of both inputs quickly reached wmax and stayed there. The same happened when both the decay term and hard bounds were imposed, i.e. both weights potentiated to the wmax. 3.3. Froemke et al. rule Without modification, the Froemke et al. rule suffers a fault – during spontaneous input spiking, the weights increase linearly without bound. Introducing a hard cap only stops this increase at that cap, and does not correct the anomalous increase. To try to counter this, we again introduced a decay term into the STDP procedure, i.e. wðt þ 1Þ ¼ wðtÞ λwðtÞ

ð27Þ

To obtain the results below, we used the parameters A þ ¼ 0:02, post A ¼ 0:01, τ þ ¼ 20 ms, τ ¼ 70 ms, τpre ¼ 78 ms s ¼ 35 ms, τ s


305

Fig. 3. Conventional STDP. Left: evolution of MPP (red) and LPP (green) weights over time. Average7 S.D. from 30 runs; Right: evolution of a running average of postsynaptic spike count 〈c〉 7 S:D:, although it is not used in the conventional STDP rule, to monitor the changes in average postsynaptic activity. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

Fig. 4. Conventional STDP. Postsynaptic voltage from 10 s before and 10 s after the onset of HFS. During the first 5 s, the first burst of HFS consisting of 5 trains of ten 400 Hz pulses is delivered, followed by a 1-min pause from 5 s on. The voltage trace is from a single run.

Fig. 5. Froemke et al. rule. Left: evolution of MPP (red) and LPP (green) weights over time. Average7 S.D. from 30 runs; Right: evolution of a running average of postsynaptic spike count 〈c〉 7 S:D:, albeit it is not used in this rule, in order to monitor the changes in average postsynaptic activity. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

Fig. 6. Froemke et al. rule. Postsynaptic voltage from 10 s before and 10 s after the onset of HFS. During the first 5 s, the first burst of HFS consisting of 5 trains of ten 400 Hz pulses is delivered, followed by a 1-min pause from 5 s on. The trace is from a single run.

λ ¼ 0:0000085. The average cell firing frequency with these parameters was 8.5 Hz (S.D. 0.020 Hz). Fig. 5 shows that the weights, after steadily increasing, slow down their increase about 20 min before HFS, behaviour we consider to be a baseline and apply the HFS to MPP. The HFS causes both pathways to experience LTP, however the tetanized MPP more greatly so. This is in partial concordance with the experiment – however, we are seeing no heterosynaptic LTD. It is also of note that this rule takes a significantly long time for the weights to arrive at more or less stable values, not occurring until well after the 60 min of stabilisation period. To be sure that

the cell in fact stabilised, we tested the cell's behaviour over time without any HFS, with the weights remaining slowly increasing without any HFS applied. Monitoring the average postsynaptic spiking reveals a temporary increase during HFS, then return to the pre-HFS level. However, potentiation of both inputs lasts, or rather continues the trend of increase after HFS. The individual postsynaptic voltage trace (Fig. 6) immediately before and after the onset of HFS does not say much, except, like in conventional STDP simulations, there seems to be more postsynaptic spikes after the onset of HFS, which is confirmed by the transient increase in activity average in Fig. 5.

306


Fig. 7. Pfister & Gerstner rule. Left: evolution of MPP (red) and LPP (green) weights over time. Average7 S.D. from 30 runs; Right: evolution of the average postsynaptic spike count 〈c〉 7 S:D:, albeit it is not used in this rule, in order to monitor the changes in average postsynaptic activity. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

Fig. 8. Pfister & Gerstner rule. Postsynaptic voltage from 10 s before and 10 s after the onset of HFS. During the first 5 s, the first burst of HFS consisting of 5 trains of ten 400 Hz pulses is delivered, followed by a 1-min pause from 5 s on. The trace is from a single run.

3.4. Pfister & Gerstner rule With the unmodified Pfister & Gerstner rule, we were unable to gain concordance with experiment for any parameter set chosen. Shown are the results for the same parameters we will use in our modified variant of this rule: A2þ ¼ 0:02, A2 ¼ 0:008, A3þ ¼ 0:0000006, A3 ¼ 0:0000003, τ þ ¼ 30 ms, τ ¼ 100 ms, τx ¼ 50 ms, τy ¼ 120 ms. The average cell firing frequency with these parameters in this STDP model was 8.2 Hz (S.D. 0.011 Hz). With this rule, HFS has no effect, with the weights diverging from the start to approach asymptotic values. When HFS is not applied, the results are identical, and testing with a very long simulation time showed the weights to remain at their stable values indefinitely (Fig. 7, left). The original Pfister & Gerstner triplet rule (2006) was not tested in a stimulation scenario, which includes simulation of ongoing spontaneous activity of neurons plus the HFS protocol. HFS in our simulations and in the real experiment consisted of very brief trains of HFS. Each train lasted only 22.5 ms. The trains were delivered in bursts of 5 trains with a 1-s pause between these 5 trains. There were 10 bursts, separated by a pause of 1 min. During all these pauses in HFS, the spontaneous input activity is still ongoing. As we now show in Fig. 7, on the right, the running average of postsynaptic spiking actually does not change with the onset and cessation of HFS. This is also supported by observing the individual trace of postsynaptic voltage (Fig. 8). 3.5. Pfister & Gerstner rule with metaplasticity After the failure of the Pfister rule to reproduce the experiment, we modified it with BCM-like metaplasticity of STDP amplitudes as in the Benuskova & Abraham rule (Eqs. (7) and (8)). To achieve these results, we use the parameters A2þ ð0Þ ¼ 0:02, A2 ð0Þ ¼ 0:008, A3þ ð0Þ ¼ 0:0000006, A3 ð0Þ ¼ 0:0000003, τ þ ¼ 30 ms, τ ¼ 100 ms, τx ¼ 50 ms, τy ¼ 120 ms. The average cell firing frequency with these parameters in this STDP model was 2.3 Hz (S.D. 0.025 Hz).

With this modification, we see a marked improvement over the original Pfister & Gerstner triplet rule – it is now in concordance with experimental data. Although the weights stabilise to different values before HFS and then “switch” when it is applied, we are still seeing homosynaptic LTP and heterosynaptic LTD, as we did with Benuskova & Abraham rule. Later, we experimented with this rule with one small adjustment – the low-frequency (0.1 Hz) asynchronous component of the spontaneous input (where either the MPP or the LPP fires but not both) was turned off. With this adjustment, the baseline weights have more or less equal values and we can still see homosynaptic LTP of the tetanised input and heterosynaptic LTD of the untetanised input (Fig. 10). Clearly under this rule, the small deviations from the synchronous input are sufficient to cause the weights to stabilise at different values during spontaneous input spiking. When inspecting the running average of the postsynaptic spike count 〈c〉 (Fig. 9, right), we see no apparent change due to HFS, perhaps just an insignificant bump. However, when we plot the postsynaptic voltage from a single run immediately before and after the onset of an HFS train (Fig. 11), we can see that during the HFS train there is a reduction in the number of postsynaptic spikes that ceases when the spontaneous input spiking resumes. This short-lived drop in the number of postsynaptic spikes during HFS is enough to observe homosynaptic LTP and heterosynaptic LTD, albeit with much lower magnitudes than in the real experiment. 3.6. Clopath et al. rule Similar to the Froemke et al. rule, the Clopath et al. rule exhibited a linear increase in the weights up to the hard bounds (which were necessary here, as without them, the weight that undergoes heterosynaptic LTD will drop below zero), and as with the former rule, here we also introduce a decay term. To obtain the results depicted below, we used the parameters A þ ¼ 0:00038, A ¼ 0:00003, τ þ ¼ 8 ms, τ ¼ 18 ms, τx ¼ 16 ms, θ þ ¼ 38, θ ¼ 41; wmin ¼ 0:01; wmax ¼ 50; c0 ¼ 5. The average cell firing frequency with these parameters in this STDP model was 8.5 Hz (S. D. 0.026 Hz). With this rule, we see an overall concordance with the experimental results as far as the gross weight changes are concerned. This rule produces a strong lasting homosynaptic LTP of the tetanised input and a weak transient heterosynaptic depression of the untetanised input, which eventually potentiates above the baseline pre-HFS value (Fig. 12, left). It is interesting that HFS actually produces an increase in the average postsynaptic spike count during HFS (Fig. 12, right). We think that this increase is due to a huge homosynaptic LTP of the tetanised input. Untetanised input manifests only weak transient depression that cannot counterbalance the overall increase in synaptic drive. The


307

Fig. 9. Pfister & Gerstner rule with metaplasticity. Left: evolution of MPP (red) and LPP (green) weights over time. Average of 30 runs 7 S.D.; Right: corresponding evolution of the running average of postsynaptic spike count 〈c〉 7 S:D:, which is now used to metaplastically adjust the triplet rule amplitudes. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

4. Discussion

Fig. 10. Pfister & Gerstner rule with metaplasticity. Evolution of MPP (red) and LPP (green) weights over time for Pfister rule with metaplasticity, with 0.1 Hz asynchronous spontaneous input removed. Average of 30 runs 7 S.D. Identical parameters to above. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

Fig. 11. Pfister & Gerstner rule with metaplasticity. Postsynaptic voltage from 10 s before and 10 s after the onset of HFS. During the first 5 s, the first burst of HFS consisting of 5 trains of ten 400 Hz pulses is delivered, followed by a 1-min pause from 5 s on. The trace is from a single run.

individual voltage trace in Fig. 13 confirms too, as there is an increase in postsynaptic spiking after the onset of HFS. However, in order to achieve these results, the integration window for the LTD metaplasticity needed to be reduced to only 1 s compared to the 60 s used in Benuskova & Abraham. Increasing the window even modestly above this level caused rapid deterioration of the results – a longer window first removes the heterosynaptic LTD, and with further increases, the post-HFS weight stability is lost. The 1-s window for integration of the mean postsynaptic voltage to adjust A was used in the original paper of Clopath et al. [22], although the authors claim that it can be longer. Later Zenke et al. [33] determined that activity in the spiking network is stable when the τ for A is from the interval (3 s, 25 s). Anyway, with a 1-s window, we only effectively consider very recent spikes (given a frequency of 8 Hz, only the last 8 or so spikes) for metaplasticity. It is possible that this quite high firing frequency of the postsynaptic neuron is behind the instability caused by longer integration windows in our simulations.

We have implemented the spiking model of the granule cell in the hippocampal dentate gyrus. Our goal was to reproduce the homosynaptic long-term potentiation of the tetanised input and heterosynaptic long-term depression of the untetanised input, as observed in real experiments after applying HFS to the MPP input [15]. Even if the model is extremely simple and abstract, e.g. it does not have dendrites, detailed voltage-gated and receptorgated membrane ionic conductances, etc., it can still reproduce experimental data quite faithfully as was previously shown in [15] and also in this report. Thus, the first area of further work could be to use a more biophysically realistic compartmental model of the granule cell with dendritic tree, like the one introduced by Aradi and Holmes [1] and used by Santhakumar et al. [32] in their computational model of dentate gyrus. The goal would be to test which synaptic learning rule works in a more biologically faithful model of the neuron. Another area of investigation is to implement more realistic spontaneous input activity and how this affects the working of the STDP rule and spike interactions. For experiments that are done in living animals one cannot omit the fact that neurons are spontaneously active and thus a good model of spontaneous activity is needed. Currently we use a Poisson spike train, which is a good first approximation, but experimental data suggest that the MPP and LPP spontaneous spiking is more akin to a quasi-periodic pattern [29,28]. For instance, Hayashi and Nonaka [31] in their computational investigation of cooperation and competition between MPP and LPP used Poisson train for LPP input and regular periodic spiking for MPP input. The data from experimental measurements show that both input activities are somewhere in between fully periodic and fully random [28]. The third gross simplification in our model is that it only has two synaptic weights, whereas in reality each input pathway has many thousands of them. Since we are building the model incrementally, first we need to understand interactions on a small scale before moving on to larger scale input simulations. The placeholder for this extension are the parameters NMPP and NLPP . At present the input spontaneous activity, albeit Poisson-like, is fully correlated between the two inputs (except for the period of HFS). This strict condition will be relaxed when moving on to hundreds of plastic synapses instead of just two. Yet another simplification is that we only consider two major excitatory input pathways, the MPP and LPP. It would be desirable to include the local and contralateral excitatory inputs, and also inhibitory circuitry similar to that shown in [32] and [31] even if only with our simple spiking model instead of a more complex compartmental one. However, it is always useful to understand

308


Fig. 12. Clopath et al. rule. Left: evolution of MPP (red) and LPP (green) weights over time for Clopath et al. rule. Average of 10 runs 7 S.D.; Right: corresponding evolution of the average postsynaptic spike count 〈c〉 7 S:D: which is used to metaplastically adjust the amplitude A (Eq. (8)). (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

Fig. 13. Clopath et al. rule. Postsynaptic voltage from 10 s before and 10 s after the onset of HFS. During the first 5 s, the first burst of HFS consisting of 5 trains of ten 400 Hz pulses is delivered, followed by a 1-min pause from 5 s on. The trace is from a single run.

what is going on without inhibition when we model plasticity of excitatory synapses. It is also worth noting that the other STDP-derived rules we consider (i.e. [20–22]) were all designed for different areas of the brain or hippocampus with different cell dynamics and input activities. A theory which produces a good result in one area of the brain may not necessarily have the same success elsewhere; hence any conclusions made regarding direct comparison of rules in a single area of the brain must take this into account. We would like to stress too, that none of these STDP-derived rules without metaplasticity was able to reproduce the average postsynaptic firing frequency of GCs during spontaneous input activity. GCs are quiet cells, spontaneously they spike seldomly, with frequencies o2 Hz [29]. However, although we make several major simplifications, we are able to observe marked differences in the evolution of synaptic weights between different STDP-derived rules and we propose that metaplasticity is an important, if not integral, component in the mechanisms of synaptic plasticity.

5. Conclusions In comparing the STDP rules we examine here, we see a range of outcomes for the evolution of the system's weights. Closest to the results from experiment are the Benuskova & Abraham, the Clopath et al. rule, and the metaplastic Pfister & Gerstner rule. Partial concordance was achieved with the Froemke et al. rule (i.e. homosynaptic LTP of the tetanized path) and a complete failure to reproduce the results for the Pfister rule and the conventional STDP rule. However given that the Pfister & Gerstner and Clopath et al. rules are both rather complicated, with a larger parameter space (and hence much more time required to optimise for a given situation, and with the increased complexity, potentially reduced performance on larger scale simulations), we conclude that for this purpose, the Benuskova & Abraham model is the superior choice of those we investigated. In addition, the Benuskova & Abraham model and metaplastic Pfister & Gerstner rule are the only ones of

those we investigated in which we were able to fully tune the firing rate of the cell. Clearly, the choice of synaptic plasticity model has a profound effect on the accuracy of the results gained from computational modelling of a neuron. The failure of conventional STDP shows that the theory of STDP alone is incapable of a complete description of synaptic plasticity. Those results that conformed closest to the experiment all featured at least some level of BCM-like metaplasticity, with Pfister rule's failure turned into a reasonable concordance when BCM-like dynamic adjustment of STDP amplitudes was added. Although the Froemke rule did not produce heterosynaptic LTD, a “metaplasticity” of sorts lies in its suppression mechanism, which likely accounts for its superior performance over conventional STDP. Given the proposed link between STDP and metaplasticity, the results we have here further support the simultaneous consideration of both theories of synaptic plasticity.

Acknowledgements We thank the reviewers for their critical and constructive comments on the manuscript. The paper was written with the support of a postgraduate publishing bursary from the University of Otago Graduate Research School.

References [1] I. Aradi, W.R. Holmes, Role of multiple calcium and calcium-dependent conductances in regulation of hippocampal dentate granule cell excitability, J. Comput. Neurosci. 6 (3) (1999) 215–235. [2] G.-Q. Bi, M.-M. Poo, Synaptic modification by correlated activity: Hebbs postulate revisited, Annu. Rev. Neurosci. 24 (2001) 139–166. [3] R.A. Nicoll, K.W. Roche, Long-term potentiation: peeling the onion, Neuropharmacology 74 (2013) 18–22. [4] M. Mayford, E.R. Kandel, Genetic approaches to memory storage, Trends Genet. 15 (1999) 463–470. [5] S. Dudek, M.F. Bear, Homosynaptic long-term depression in area CAl of hippocampus and effects of N-methyl-D-aspartate receptor blockade, Proc. Natl. Acad. Sci. USA 89 (1992) 4363–4367. [6] W.C. Abraham, S.E. Mason-Parker, B. Logan, Low-frequency stimulation does not readily cause long-term depression or depotentiation in the dentate gyrus of awake rats, Brain Res. 722 (1996) 217–221. [7] D. Habib, H.C. Dringenberg, Alternating low frequency stimulation of medial septal and commissural fibers induces NMDA-dependent, long-lasting potentiation of hippocampal synapses in urethane-anesthetized rats, Hippocampus 19 (2009) 299–307. [8] E.L. Bienenstock, L.N. Cooper, P.W. Munro, Theory for the development of neuron selectivity: orientation specificity and binocular interaction in visual cortex, J. Neurosci. 2 (1982) 32–48. [9] L. Benuskova, F.F. Ebner, M.E. Diamond, M. Armstrong-James, Computational study of experience-dependent plasticity in adult rat cortical barrel-column, Network: Comput. Neural Syst. 10 (4) (1999) 303–323.


[10] L. Benuskova, R. Velayudhan, M. Armstrong-James, E.F. Ebner, Theory for normal and impaired experience-dependent plasticity in neocortex of adult rats, Proc. Natl. Acad. Sci. USA 98 (2001) 2797–2802. [11] W.C. Abraham, Metaplasticity: tuning synapses and networks for plasticity, Nat. Rev. Neurosci. 9 (2008) 387–401. [12] R. Krueppel, S. Remy, H. Beck, Dendritic integration in hippocampal dentate gyrus granule cells, Neuron 71 (2011) 512–528. [13] H. Markram, W. Gerstner, P.J. Sjöström, A history of spike-timing-dependent plasticity, Front. Syn. Neurosci. 3 (2011) 4, http://dx.doi.org/10.3389/ fnsyn.2011.00004. [14] C.G. Mayr, J. Partzsch, Rate and pulse based plasticity governed by local synaptic state variables, Front. Syn. Neurosci. 2 (2010) 33, http://dx.doi.org/ 10.3389/fnsyn.2010.00033. [15] L. Benuskova, W.C. Abraham, STDP rule endowed with the BCM sliding threshold accounts for hippocampal heterosynaptic plasticity, J. Comput. Neurosci. 22 (2007) 129–133. [16] W.C. Abraham, S.E. Mason-Parker, M.F. Bear, S. Webb, W.P. Tate, Heterosynaptic metaplasticity in the hippocampus in vivo: a BCM-like modifiable threshold for LTP, Proc. Natl. Acad. Sci. USA 98 (2001) 10924–10929. [17] E.M. Izhikevich, N.S. Desai, Relating STDP to BCM, Neural Comput. 15 (2003) 1511–1552. [18] P.J. Sjöström, G.G. Turrigiano, S.B. Nelson, Rate, timing, and cooperativity jointly determine cortical synaptic plasticity, Neuron 32 (2001) 1149–1164. [19] Y.-W. Lin, H.-W. Yang, H.-J. Wang, C.-L. Gong, T.-H. Chiu, M.-Y. Min, Spiketiming dependent plasticity at resting and conditioned lateral perforant path synapses on granule cells in the dentate gyrus: different roles of N-methyl-Daspartate and group I metabotropic glutamate receptors, Eur. J. Neurosci. 23 (2006) 2362–2374. [20] R.C. Froemke, I.A. Tsay, M. Raad, J.D. Long, Y. Dan, Contribution of individual spikes in burst-induced long-term synaptic modification, J. Neurophysiol. 95 (3) (2006) 1620–1629. [21] J.P. Pfister, W. Gerstner, Triplets of spikes in a model of spike timingdependent plasticity, J. Neurosci. 26 (38) (2006) 9673–9682. [22] C. Clopath, L. Bs̈ing, E. Vasilaki, W. Gerstner, Connectivity reflects coding: a model of voltage-based STDP with homeostasis, Nat. Neurosci. 13 (3) (2010) 344–352. [23] A. Morrison, M. Diesmann, W. Gerstner, Phenomenological models of synaptic plasticity based on spike timing, Biol. Cybern. 98 (6) (2008) 459–478. [24] B.L. Murphy, R.Y.K. Pun, H. Yin, C.R. Faulkner, A.W. Loepke, S.C. Danzer, Heterogenous integration of adult-generated granule cells into the epileptic brain, J. Neurosci. 31 (1) (2011) 105–117. [25] E. Förster, S. Zhao, M. Frotscher, Laminating the hippocampus, Nat. Rev. Neurosci. 7 (2006) 259–268. [26] E.M. Izhikevich, Simple model of spiking neurons, IEEE Trans. Neural Netw. 14 (2003) 1569–1572. [27] M. Rudolph, A. Destexhe, The discharge variability of neocortical neurons during high conductance states, Neuroscience 119 (2003) 855–873. [28] S.S. Deshmukh, D. Yoganarasimha, H. Voicu, J.J. Knierim, Theta modulation in the medial and the lateral entorhinal cortices, J. Neurophysiol. 104 (2010) 994–1006.

309

[29] A. Kimura, C. Pavlides, Long-term potentiationdepotentiation are accompanied by complex changes in spontaneous unit activity in the hippocampus, J. Neurophysiol. 84 (2000) 1894–1906. [30] E.E. Clothiaux, M. Bear, L.N. Cooper, Synaptic plasticity in visual cortex: comparison of theory with experiment, J. Neurophysiol. 66 (1991) 1785–1804. [31] H. Hayashi, Y. Nonaka, Cooperation and competition between lateral and medial perforant path synapses in the dentate gyrus, Neural Netw. 24 (2011) 233–246. [32] V. Santhakumar, I. Aradi, I. Soltesz, Role of mossy fiber sprouting and mossy cell loss in hyperexcitability: a network model of the dentate gyrus incorporating cell types and axonal topography, J. Neurophysiol. 93 (1) (2005) 437–453. [33] F. Zenke, G. Hennequin, W. Gerstner, Synaptic plasticity in neural networks needs homeostasis with a fast rate detector, PLoS Comput. Biol. 9 (11) (2013) e1003330, http://dx.doi.org/10.1371/journal.pcbi.1003330.

Nicholas Hananeia is a postgraduate student in the Department of Computer Science at the University of Otago, Dunedin, New Zealand. He has background in physics and mathematics. He has then pursued postgraduate research at the Master's level in computational neuroscience, specifically in the field of synaptic plasticity modelling.

Lubica Benuskova is an Associate Professor at the Department of Computer Science at the University of Otago. Her research interests include neurobiologically inspired computing and dynamics of biological and artificial neural networks. Specific topics of interest include biological learning rules, spiking neural networks, classification and prediction by means of neural networks and computational neurogenetic modelling (investigating the dynamic influence of internal gene regulatory networks upon neural dynamics).