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Triplet-based spike timing dependent plasticity (TSTDP) modeling using VHDL-AMS
Neurocomputing 149 (2015) 1440–1444
Contents lists available at ScienceDirect
Neurocomputing journal homepage: www.elsevier.com/locate/neucom
Letters
Triplet-based spike timing dependent plasticity (TSTDP) modeling using VHDL-AMS Afsaneh Shahim-Aeen, Gholamreza Karimi n Department of Electrical Engineering, Faculty of Engineering, Razi University, Kermanshah 67149, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 10 December 2013 Received in revised form 17 August 2014 Accepted 19 August 2014 Communicated by R. W. Newcomb Available online 1 September 2014
Spike Timing Dependent Plasticity is one of the synaptic plasticity rules that plays an important role in learning and memory in the brain. There are two rules to describe STDP; the conventional one is pairbased STDP (PSTDP) and the other one is triplet-based STDP (TSTDP) that is a powerful synaptic plasticity rule and acts beyond the classical rule. Although PSTDP fails to reproduce some of the experimental observations, TSTDP is capable of reproducing them. In this paper, a VHDL-AMS based TSTDP model is presented which exhibits the behavioral model of triplet-based spike timing dependent plasticity. The proposed model is simulated using Ansoft Simplorer. This model has similar results to the mentioned experimental observations and is capable of being employed in different analog or digital implementations of neuromorphic systems. & 2014 Elsevier B.V. All rights reserved.
Keywords: Synaptic plasticity Spike timing dependent plasticity (STDP) Rate-based plasticity Neuromorphic engineering VHDL-AMS
1. Introduction
2. Synaptic plasticity
The ability of a synaptic connection to change in strength is called synaptic plasticity and is responsible for learning and memory in the brain [1]. The strength of the connection between two neurons can vary via spike-timing, varying the frequency of inputs to the neuron or changes to the internal concentration of calcium in the neuron's spine apparatus [2–7]. For many years, engineers have been trying to implement different neuron models and learning rules in analog or digital electronic circuits in order to have physical realization of them [8–15]. Due to non-linearity of the mathematical representations of the biological systems, their implementations need some approximations. As a result, VHDL-AMS can be very useful for solving the problem. As VHDL-AMS can be used for behavioral modeling, nonlinear systems can be modeled similar to linear systems without using approximations [16,17]. Furthermore, by using VHDL-AMS modeling, digital and analog parts of a system can be mixed together. Our goal in this paper is to describe the processes of synaptic plasticity leading to learning and memory using VHDL-AMS. The rest of this paper is organized as follows: in Section 2, the synaptic plasticity and its rules are described. A description of the proposed model and simulation results are given in Sections 3 and 4 respectively, and Section 5 provides the conclusion.
As synapses have plastic property either their shapes or their functions can change over periods of time that could last for a few seconds, a few minutes, a few hours, or perhaps even a lifetime. Changing the structure or function of the synaptic connections leads to learning and memory. As indicated in [18], Fig. 1 illustrates the synaptic action. In the left part of this figure (a), the connection between a pre-synaptic neuron and a post-synaptic neuron named a synapse is shown, and in the right part (b), the details of a synaptic junction are demonstrated. Synaptic strength determines the effect of a pre-synaptic spike contributing to the cumulative action at the post-synaptic neuron. The synaptic strength changes in time as a function of the spiking activity of pre- and postsynaptic neurons. Researchers have shown that the timing and rate of spikes can change the strength of the synapses. The spike timing dependent plasticity (STDP) and the Bienenstosk Cooper Munro (BCM) rules are two well-known synaptic plasticity rules that are time-based and rate-based, respectively [4,5]. BCM rule is a rate-based rule that is based on the fact that the rate of the presynaptic stimulation can modulate the synaptic strength. As one of the important characteristics of each spike train is the rate of spikes, many researchers believe that it causes synaptic plasticity [5]. It is believed that the STDP rule has an important role in learning and memory in the brain. To discovery of STDP, several models were proposed and developed to explain it. These models aim to reproduce the experimental data with a mathematical expression. Its classical description is the
n
Corresponding author.
http://dx.doi.org/10.1016/j.neucom.2014.08.050 0925-2312/& 2014 Elsevier B.V. All rights reserved.
A. Shahim-Aeen, G. Karimi / Neurocomputing 149 (2015) 1440–1444
1441
potentiation and depression contributions sum linearly in PSTDP. Another disadvantage of PSTDP is about BCM rule which is described in previous section. According to this rule, synaptic plasticity depends linearly on the pre synaptic and non-linearly on the post synaptic activities [5]. It has been shown that TSTDP can exhibit the analogous behavior to the BCM rule. As mentioned above, TSTDP is a very powerful synaptic plasticity rule and because of this, implementation of this rule is more useful than the others. In the following sections, the VHDL-AMS model of this rule and its simulation results are presented.
3. Modeling TSTDP using VHDL–AMS
Fig. 1. Demonstration of synaptic action (a) connection between a pre-synaptic neuron and a post-synaptic neuron, (b) details of a synaptic junction [18].
pair-based STDP (PSTDP) which describes the STDP responses as two separate levels of synaptic plasticity subject that decays exponentially. Eq. (1) is the mathematical representation of this rule [19]: 8 þ > if Δt 4 0 < Δω þ ¼ þA exp τΔtþ Δω ¼ ð1Þ Δt > if Δt r 0 : Δω ¼ A exp τ where Δω denotes the weight change of the synapse. Δt ¼ t post t post where t post is the arrival time of the postsynaptic input and t pre is the arrival time of the presynaptic input, and A is the amplitude of the weight change that has a positive value. So, A þ , A denote potentiation and depression amplitude parameters respectively that have different positive values. According to this model, the synaptic plasticity is characterized as a function of the time difference between pairs of spikes. Another description of STDP is the triplet-based spike timing dependent plasticity (TSTDP) which considers sets of three spikes, i.e., two pre and one post or one pre and two post [2]. Its mathematical representation is defined by Eq. (2): 8 Δt þ þ Δt > < Δω þ ¼ exp τ þ 1 A2 þA3 exp τy 2 Δω ¼ ð2Þ 1 3 > A2 þ A3 exp τΔt : Δω ¼ exp Δt τ y where Δω ¼ Δω þ for t ¼ t post and if t ¼ t pre then the weight change is Δω ¼ Δω . A2þ , A3þ ; A2 and A3 denote potentiation and depression amplitude parameters respectively. Δt 1 ð ¼ t postðnÞ t preðnÞ Þ, Δt 2 ð ¼ t postðnÞ t postðn 1Þ εÞ and Δt 3 ð ¼ t preðnÞ t preðn 1Þ εÞ are the time differences between combinations of pre and post synaptic spikes, and ε is a small positive constant to ensure that the weight update selects the correct values occurring just before the pre- or post-synaptic spike of interest [20]. τ , τ þ , τx and τy are time constants. Before this rule, there was another rule proposed in [4] which considers quadruplet of spikes to induce synaptic modification. Simulation results of TSTDP rule show that this rule can explore the impact of quadruplet combination of spike patterns on synaptic plasticity. TSTDP rule has two models. The first one is the full triplet model and all of the parameters of Eq. (2) are taken into account for it. The second one is the minimal model which disregardsA3 [2].The PSTDP has some deficiencies such as being unable to reproduce all of the experimental results, but TSTDP can reproduce them. As it was reported in [4], the interactions between spikes in higher order spike patterns are not linear but the
As it is known, the first step in integrated circuit design is the system specification (high-level specifications) where the initial idea is defined. Next step is description of the circuit at architectural level, where it is partitioned into functional blocks. This includes descriptions using proper hardware description languages (HDLs) like VHDL-AMS. The last step in the design flow is defined at circuit level. But at circuit level, the circuit simulators are not capable of handling efficiently larger designs at lower levels (circuit level) of abstraction due to its computational intensive nature. An alternate approach is to capture the behavior of the AMS designs at higher level of abstraction using hardware description languages like VHDL-AMS. This technique (behavioral modeling) brings down the simulation time of the design while having good accuracy and run-time. Thus, the designers can look at modeling AMS designs at higher levels of abstraction, providing an attractive platform to carry out such modeling in the low level (circuit level) [21]. As a result, VHDL-AMS can be used for behavioral modeling so that it can exhibit the realistic behavior of the system. Furthermore, most of the mathematical representations of the biological systems are non-linear and need to have linear approximations for being implemented. So, by using the VHDL-AMS for modeling these equations, there is no need to approximate them, while observing the exact simulation results of them. The VHDL-AMS modeling of TSTDP has been done as follows. We have modeled the behavior of the TSTDP as a component that has two ports which are defined as electrical ports, so, two inputs are required; one represents a glutamate signal that carries information about presynaptic stimuli and the other represents a dendritic action potential signal that carries information about the post synaptic stimuli. In other words, one presynaptic input and one postsynaptic input are needed. We modeled these two inputs with two pulses whose delay times and frequencies can be adjusted in the test bench according to what is needed for simulation. For example, to have the exponential learning window based on the pairing protocol, we need two pulses with appropriate delay times that are repeated at each second. By sweeping their delay times, various time differences will be obtained. These input pulses are connected to the two ports of TSTDP component which were defined above, so, in the architecture of this component, the arrival times of postsynaptic pulse and presynaptic pulse are determined and the weight change is calculated in both minimal and full models. All reported experiments in this paper assume the nearest spike interaction, which considers the interaction of a spike only with its two immediate succeeding and preceding nearest neighbors. The values of the amplitudes of the weight changes are the reported data of [2] that are given in Tables 1 and 2. As reported in [2], these parameters are set in such a way that the normalized mean square error across all experimental protocols is minimal. The additional parameters τ þ ¼ 16:8 ms and τ ¼ 33:7 ms are taken from [7]. A block diagram of our model is shown in Fig. 2. As described above, triplet learning rules can reproduce triplet and quadruplet
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Table 1 List of the parameters used to model the visual cortex data set as reported in [2]. Parameter
A2þ A3þ A2 A3 τ x ðmsÞ τ y ðmsÞ
Model Full
Minimal
8:8 10 11
0
5:3 10 2
5 10 2
6:6 10 3
8 10 3 0
3:1 10 714 40
3
– 40
Table 2 List of the parameters used to model the hippocampal culture data set as reported in [2]. Parameter
A2þ A3þ A2 A3 τ x ðmsÞ τ y ðmsÞ
Model Full
Minimal
5:3 10 3
4:6 10 3
9:1 10 3
9:1 10 3
3 10
3
7:5 10 9 575 47
3 10 3 0 – 48
experiments. They can also reproduce frequency effects, so we tested our model with these effects too. The required experimental protocols are pairing protocol, triplet protocol and quadruplet protocol. In the pair-based STDP protocol, 60 pairs of pre and post synaptic spikes with a delay of Δt ¼ t post t pre and repetition frequency of 1 Hz are transmitted [1,7]. For triplet experiments, two patterns of spikes that consist of 60 triplets of spikes with repetition frequency of 1 Hz are used. One of these patterns is composed of one post synaptic and two pre synaptic spikes in a pre-post-pre configuration and the other is composed of two post synaptic and one pre synaptic spikes in a post-pre-post configuration. In the first one, the timing differences are defined as Δt1 ¼ t post t pre1 and Δt2 ¼ t post t pre2 , where t pre1 and t pre2 denote the first and second presynaptic spikes of the triplet. In the second one, they are defined as Δt ¼ t post1 t pre and Δt2 ¼ t post2 t pre , where t post1 and t post2 are the first and second postsynaptic spikes of the triplet respectively. Also, quadruplet protocol consists of 60 quadruplets of spikes repeated at a given frequency of 1 Hz. This protocol is composed of either a post–pre pair with a delay of Δt1 ¼ t post1 t pre1 o 0, or pre–post pair with a delay of Δt2 ¼ t post2 t pre2 4 0. In this protocol, the time T is defined by T ¼ ððt post2 þ t pre2 Þ=2Þ ððt post1 þ t prt1 Þ=2Þ, when the post-pre pair precedes the pre-post pair, T is positive and when the opposite happens, T is negative. Identical to Ref. [2], in all quadruplet experiments in this paper, we took Δt ¼ Δt1 ¼ Δt2 ¼ 5 ms.
4. Simulation results The results of TSTDP simulation with VHDL-AMS in Ansoft Simplorer are given in Figs. 3 to 6. The simulation results of [2] are also inserted into these figures to show the close fit of our model to what was reported in this original triplet article that had an optimization scale with the highest analogy to the experimental data. To test our model on a consistent set of data, we took the measurements of [6,7] as are reported in Tables 3 and 4. These available experimental data are inserted to these figures. Fig. 3 shows that the proposed
Fig. 2. The block diagram of our model.
model can reproduce the exponential learning window produced by both minimal and full TSTDP models under the conventional pairing protocol. As it is shown, when the postsynaptic spike precedes the presynaptic one (when Δt ¼ t post t pre is more than zero) the potentiation of the synapse occurs, i.e., Δw 4 0. The depression of the synapse or Δw o 0 is occurred when Δt ¼ t post t pre is less than zero. By increasing the absolute values of Δt, the absolute values of Δw decreased. The vertical bars shown in this figure are the available experimental data reported in Table 4 for pairing protocol which are at Δt ¼ 10 ms and Δt ¼ 10 ms. Fig. 4 depicts that this model succeeds to reproduce the triplet experiments with its minimal and full models. In this figure the values of our results and the experimental data, which are reported in Table 4, in pre-post-pre and post-pre-post combinations of spikes for four different values of Δt1 and Δt2 are shown. In Fig. 5 the simulation results of the minimal and full models for the quadruplet experiments are illustrated. This figure shows the weight change of the synapse as a function of T for Δt ¼ 5 ms. For both T40 and To0, the value of Δw decreases when T is increased. Similar to Fig. 3, in this figure the experimental data of the quadruplet
A. Shahim-Aeen, G. Karimi / Neurocomputing 149 (2015) 1440–1444
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Fig. 3. Pairing protocol produced by the proposed model. (a) The weight change in minimal model, (b) The weight change in full model. The required parameters are taken from Table 2.
Fig. 4. Triplet protocol produced by the proposed model.(a) the weight change in minimal model of post-pre-post combination of spikes, (b) the weight change in full model of post-pre-post combination of spikes, (c) the weight change in minimal model of pre-post-pre combination of spikes, (d) the weight change in full model of pre-post-pre combination of spikes. The required parameters are taken from Table 2.
Fig. 5. Quadruplet protocol produced by the proposed model. (a) The weight change in minimal model,(b) the weight change in full model. The required parameters are taken from Table 2.
Fig. 6. Weight change in pairing protocol as a function of pairing frequency ρ reproduced by the proposed model.(a) the weight change in minimal model, (b) the weight change in full model. The required parameters are taken from Table 1.
protocol for 3 values of T( 88.5 ms, 83.7 ms, and 20 ms), as indicated in Table 4, are inserted as vertical bars. Finally, as Fig. 6 shows, these proposed minimal and full models have a BCM like behavior for the
positive and negative values ofΔt. The experimental data of the BCM rule, according to what is indicated in Table 3 (Δt ¼ 10 ms and Δt ¼ 10 ms ), are inserted into this figure to show their good
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Table 3 Experimental weight change as a function of the delay and the rate of spikes in pairing protocol as reported in [6]. ρðHzÞ
Δt ¼ 10 ms Δw
Δt ¼ 10 ms Δw
0.1 10 20 40 50
0.04 70.05 0.14 70.1 0.29 70.14 0.53 70.11 0.56 70.26
0.29 7 0.08 0.417 0.11 0.34 7 0.1 0.56 7 0.32 0.75 7 ;0.19
Table 4 Experimental weight change as a function of the delays in pairing, quadruplet and triplet protocols as reported in [7]. Pairing
Δt 10 ms 10 ms
Δw 0:25 7 0:05 0:177 0:05
Quadruplet Δt ¼ 5 ms
TðmsÞ 88:5 83:7 20
Δw 0:003 7 0:03 0:06 7 0:04 0:21 7 0:04
Triplet (2-pre,1-post)
Δt1 5 10 15 5
Δt2 5 10 5 15
Δw 0:017 0:04 0:03 7 0:04 0:01 7 0:03 0:21 7 0:04
Triplet (1-pre,2-post)
Δt1 5 10 15 5
Δt2 5 10 5 15
Δw 0:33 7 0:04 0:34 7 0:04 0:22 7 0:08 0:29 7 0:05
matching. For validation of the proposed model, some different patterns of spikes including spike pairs, spike triplets and spike quadruplets were used. As it is obvious from these figures, the proposed model has similar results to the mentioned experimental observations. It can be seen that all the results are approximately the same as the results reported in [2]. In order to test the proposed model under the mentioned protocols, full TSTDP and minimal TSTDP rules are used, and it has been demonstrated that the minimal model can reproduce very similar results to those of the full TSTDP model. 5. Conclusion This paper presented a new modeling of TSTDP using VHDL-AMS with Ansoft Simulator. As behavioral simulations show the major characteristics of the nonlinear system such as biological systems, design errors are detected early and predicted. The goal of behavioral modeling in higher level building blocks is to ease and accelerate the design of analog and mixed circuits, reducing simulation time without linear approximations, being implemented. The proposed model is a synapse that can be employed in various neuromorphic systems that are implemented in analog or digital circuits, aiming for different applications and having physical realization of them. This helps the designers to use an efficient methodology to realize physical implementation in neuromorphic VLSI circuits. References [1] G. Bi., M. Poo, Synaptic modifications in cultured hippocampal neurons: dependence on spike timing, synaptic strength, and postsynaptic cell type, J. Neurosci. 18 (24) (1998) 10464–10472.
[2] J. Pfister., W. Gerstner, Triplets of spikes in a model of spike timing dependent plasticity, J. Neurosci. 26 (38) (2006) 9673–9682. [3] Y., Meng., K., Zhou., J., Monzon, C., Poon. Iono–neuromorphic implementation of spike- timing dependent synaptic plasticity, Annual international conference of the IEEE Engineering in Medicine and Biology Society (EMBC-2011), pp. 7274–7277. [4] R. Froemke., Y. Dan, Spike-timing-dependent synaptic modification induced by natural spike trains, Nature 416 (6879) (2002) 433–438. [5] E. Bienenstock., L. Cooper., P. Munro, Theory for the development of neuron selectivity: orientation specificity and binocular interaction in visual cortex, J. Neurosci. 2 (1) (1982) 32. [6] P. Sjostrom., G. Turrigiano., S. Nelson., Rate, timing, and cooperativity jointly determine cortical synaptic plasticity, Neuron 32 (6) (2001) 1149–1164. [7] H. Wang., R. Gerkin., D. Nauen., G. Bi., Coactivation and timing dependent integration of synaptic potentiation and depression, Nat. Neurosci. 8 (2) (2005) 187–193. [8] S. Bamford., A. Murray., D. Willshaw., Spike-timing dependent plasticity with weight dependence evoked from hysical constraints, IEEE Trans. Biomed.. Circuits Syst. 6 (4) (2012) 385–398. [9] C. Mead, Analog VLSI and neural systems, Addison-Wesely Longman Publication Co., Inc., Boston, MA, USA, 1989. [10] E. Farquhar., P. Hasler., A bio-physically inspired silicon neuron, IEEE Trans. Circuits Syst. I: Regul. Pap. 52 (3) (2005) 477–488. [11] G. Indiveri., E. Chicca., R. Douglas., A VLSI array of low-power spiking neurons and bistable synapses with spike-timing dependent plasticity, IEEE Trans. Neural Netw. 17 (1) (2006) 211–221. [12] G., Indiveri., B., Linares-Barranco., T., Hamilton., A., Van Schaik., R., EtienneCummings., T., Delbruck., et al. Neuromorphic silicon neuron circuits. Frontiers in Neuroscience, 5, Art 2011, No. 73. [13] G. Rachmuth., H. Shouval., M. Bear., C. Poon., A biophysically based neuromorphic model of spike rate-and timing-dependent plasticity, Proc. Natl. Acad. Sci. 108 (49) (2011) E1266–E1274. [14] S. Davies., A. Rast., F. Galluppi., S. Furber., A forecast-based STDP rule suitable for neuromorphic implementation, Neural Netw. 32 (2012) 3–14. [15] A. Rast., F. Galluppi., S. Davies., L. Plana., C. Patterson., T. Sharp., et al., Concurrent heterogeneous neural model simulation on real-time neuromimetic hardware, Neural Netw. 24 (2011) (2011) 961–978. [16] IEEE Standard VHDL Analog and Mixed-Signal Extensions, IEEE Standard 1076.1–1999, December 1999. [17] P.J. Ashenden., G.D. Peterson., D.A. Teegarden, The System Designer's Guide to VHDL-AMS, Morgan Kaufmann, San Francisco, 2002. [18] T. Serrano-Gotarredona, T. Prodromakis, B. Linares-Barranco., A proposal for hybrid memristor-CMOS spiking neuromorphic learning systems, Circuits Syst. Mag. IEEE 13 (2013) 74–88. [19] S. Song., K. Miller., L. Abbott, Competitive hebbian learning through spiketiming-dependent synaptic plasticity, Nat. Neurosci. 3 (2000) 919–926. [20] M. Rahimi Azghadi, S. Al-Sarawi, D. Abbott, N. Iannella, A neuromorphic VLSI design for spike timing and rate based synaptic plasticity, Neural Netw. 45 (2013) 70–82. [21] R. Narayanan, N. Abbasi, G. Al Sammane, M. Zaki, S. Tahar, A Comparative Study of AMS Circuit Simulation in VHDL-AMS and SystemC-AMS, International Conference on Embedded Systems & Critical Applications (ICESA'08), 2008, pp. 23–28.
Afsaneh Shahimaeen was born in Shiraz, Iran, in 1983. She received the B.Sc. degree in electronics engineering from Razi University, Kermanshah, Iran, in 2006, where she is currently working toward the M.Sc degree in Electronic Engineering at Razi University, Kermanshah. Her current research interests include analog circuits design, artificial intelligence and neural networks.
Gholamreza Karimi was born in Kermanshah, Iran in 1977. He received the B.S. and M.S. and PhD degrees in electrical engineering from Iran University of Science and Technology (IUST) in 1999, 2001 and 2006 respectively. He is currently an Assistant Professor in Electrical Department at Razi University, Kermanshah, since 2007. His research interests include low power analog and digital IC design, RF IC design, modeling and simulation of RF mixed signal IC and microwave devices and artificial intelligence systems.
Contents lists available at ScienceDirect
Neurocomputing journal homepage: www.elsevier.com/locate/neucom
Letters
Triplet-based spike timing dependent plasticity (TSTDP) modeling using VHDL-AMS Afsaneh Shahim-Aeen, Gholamreza Karimi n Department of Electrical Engineering, Faculty of Engineering, Razi University, Kermanshah 67149, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 10 December 2013 Received in revised form 17 August 2014 Accepted 19 August 2014 Communicated by R. W. Newcomb Available online 1 September 2014
Spike Timing Dependent Plasticity is one of the synaptic plasticity rules that plays an important role in learning and memory in the brain. There are two rules to describe STDP; the conventional one is pairbased STDP (PSTDP) and the other one is triplet-based STDP (TSTDP) that is a powerful synaptic plasticity rule and acts beyond the classical rule. Although PSTDP fails to reproduce some of the experimental observations, TSTDP is capable of reproducing them. In this paper, a VHDL-AMS based TSTDP model is presented which exhibits the behavioral model of triplet-based spike timing dependent plasticity. The proposed model is simulated using Ansoft Simplorer. This model has similar results to the mentioned experimental observations and is capable of being employed in different analog or digital implementations of neuromorphic systems. & 2014 Elsevier B.V. All rights reserved.
Keywords: Synaptic plasticity Spike timing dependent plasticity (STDP) Rate-based plasticity Neuromorphic engineering VHDL-AMS
1. Introduction
2. Synaptic plasticity
The ability of a synaptic connection to change in strength is called synaptic plasticity and is responsible for learning and memory in the brain [1]. The strength of the connection between two neurons can vary via spike-timing, varying the frequency of inputs to the neuron or changes to the internal concentration of calcium in the neuron's spine apparatus [2–7]. For many years, engineers have been trying to implement different neuron models and learning rules in analog or digital electronic circuits in order to have physical realization of them [8–15]. Due to non-linearity of the mathematical representations of the biological systems, their implementations need some approximations. As a result, VHDL-AMS can be very useful for solving the problem. As VHDL-AMS can be used for behavioral modeling, nonlinear systems can be modeled similar to linear systems without using approximations [16,17]. Furthermore, by using VHDL-AMS modeling, digital and analog parts of a system can be mixed together. Our goal in this paper is to describe the processes of synaptic plasticity leading to learning and memory using VHDL-AMS. The rest of this paper is organized as follows: in Section 2, the synaptic plasticity and its rules are described. A description of the proposed model and simulation results are given in Sections 3 and 4 respectively, and Section 5 provides the conclusion.
As synapses have plastic property either their shapes or their functions can change over periods of time that could last for a few seconds, a few minutes, a few hours, or perhaps even a lifetime. Changing the structure or function of the synaptic connections leads to learning and memory. As indicated in [18], Fig. 1 illustrates the synaptic action. In the left part of this figure (a), the connection between a pre-synaptic neuron and a post-synaptic neuron named a synapse is shown, and in the right part (b), the details of a synaptic junction are demonstrated. Synaptic strength determines the effect of a pre-synaptic spike contributing to the cumulative action at the post-synaptic neuron. The synaptic strength changes in time as a function of the spiking activity of pre- and postsynaptic neurons. Researchers have shown that the timing and rate of spikes can change the strength of the synapses. The spike timing dependent plasticity (STDP) and the Bienenstosk Cooper Munro (BCM) rules are two well-known synaptic plasticity rules that are time-based and rate-based, respectively [4,5]. BCM rule is a rate-based rule that is based on the fact that the rate of the presynaptic stimulation can modulate the synaptic strength. As one of the important characteristics of each spike train is the rate of spikes, many researchers believe that it causes synaptic plasticity [5]. It is believed that the STDP rule has an important role in learning and memory in the brain. To discovery of STDP, several models were proposed and developed to explain it. These models aim to reproduce the experimental data with a mathematical expression. Its classical description is the
n
Corresponding author.
http://dx.doi.org/10.1016/j.neucom.2014.08.050 0925-2312/& 2014 Elsevier B.V. All rights reserved.
A. Shahim-Aeen, G. Karimi / Neurocomputing 149 (2015) 1440–1444
1441
potentiation and depression contributions sum linearly in PSTDP. Another disadvantage of PSTDP is about BCM rule which is described in previous section. According to this rule, synaptic plasticity depends linearly on the pre synaptic and non-linearly on the post synaptic activities [5]. It has been shown that TSTDP can exhibit the analogous behavior to the BCM rule. As mentioned above, TSTDP is a very powerful synaptic plasticity rule and because of this, implementation of this rule is more useful than the others. In the following sections, the VHDL-AMS model of this rule and its simulation results are presented.
3. Modeling TSTDP using VHDL–AMS
Fig. 1. Demonstration of synaptic action (a) connection between a pre-synaptic neuron and a post-synaptic neuron, (b) details of a synaptic junction [18].
pair-based STDP (PSTDP) which describes the STDP responses as two separate levels of synaptic plasticity subject that decays exponentially. Eq. (1) is the mathematical representation of this rule [19]: 8 þ > if Δt 4 0 < Δω þ ¼ þA exp τΔtþ Δω ¼ ð1Þ Δt > if Δt r 0 : Δω ¼ A exp τ where Δω denotes the weight change of the synapse. Δt ¼ t post t post where t post is the arrival time of the postsynaptic input and t pre is the arrival time of the presynaptic input, and A is the amplitude of the weight change that has a positive value. So, A þ , A denote potentiation and depression amplitude parameters respectively that have different positive values. According to this model, the synaptic plasticity is characterized as a function of the time difference between pairs of spikes. Another description of STDP is the triplet-based spike timing dependent plasticity (TSTDP) which considers sets of three spikes, i.e., two pre and one post or one pre and two post [2]. Its mathematical representation is defined by Eq. (2): 8 Δt þ þ Δt > < Δω þ ¼ exp τ þ 1 A2 þA3 exp τy 2 Δω ¼ ð2Þ 1 3 > A2 þ A3 exp τΔt : Δω ¼ exp Δt τ y where Δω ¼ Δω þ for t ¼ t post and if t ¼ t pre then the weight change is Δω ¼ Δω . A2þ , A3þ ; A2 and A3 denote potentiation and depression amplitude parameters respectively. Δt 1 ð ¼ t postðnÞ t preðnÞ Þ, Δt 2 ð ¼ t postðnÞ t postðn 1Þ εÞ and Δt 3 ð ¼ t preðnÞ t preðn 1Þ εÞ are the time differences between combinations of pre and post synaptic spikes, and ε is a small positive constant to ensure that the weight update selects the correct values occurring just before the pre- or post-synaptic spike of interest [20]. τ , τ þ , τx and τy are time constants. Before this rule, there was another rule proposed in [4] which considers quadruplet of spikes to induce synaptic modification. Simulation results of TSTDP rule show that this rule can explore the impact of quadruplet combination of spike patterns on synaptic plasticity. TSTDP rule has two models. The first one is the full triplet model and all of the parameters of Eq. (2) are taken into account for it. The second one is the minimal model which disregardsA3 [2].The PSTDP has some deficiencies such as being unable to reproduce all of the experimental results, but TSTDP can reproduce them. As it was reported in [4], the interactions between spikes in higher order spike patterns are not linear but the
As it is known, the first step in integrated circuit design is the system specification (high-level specifications) where the initial idea is defined. Next step is description of the circuit at architectural level, where it is partitioned into functional blocks. This includes descriptions using proper hardware description languages (HDLs) like VHDL-AMS. The last step in the design flow is defined at circuit level. But at circuit level, the circuit simulators are not capable of handling efficiently larger designs at lower levels (circuit level) of abstraction due to its computational intensive nature. An alternate approach is to capture the behavior of the AMS designs at higher level of abstraction using hardware description languages like VHDL-AMS. This technique (behavioral modeling) brings down the simulation time of the design while having good accuracy and run-time. Thus, the designers can look at modeling AMS designs at higher levels of abstraction, providing an attractive platform to carry out such modeling in the low level (circuit level) [21]. As a result, VHDL-AMS can be used for behavioral modeling so that it can exhibit the realistic behavior of the system. Furthermore, most of the mathematical representations of the biological systems are non-linear and need to have linear approximations for being implemented. So, by using the VHDL-AMS for modeling these equations, there is no need to approximate them, while observing the exact simulation results of them. The VHDL-AMS modeling of TSTDP has been done as follows. We have modeled the behavior of the TSTDP as a component that has two ports which are defined as electrical ports, so, two inputs are required; one represents a glutamate signal that carries information about presynaptic stimuli and the other represents a dendritic action potential signal that carries information about the post synaptic stimuli. In other words, one presynaptic input and one postsynaptic input are needed. We modeled these two inputs with two pulses whose delay times and frequencies can be adjusted in the test bench according to what is needed for simulation. For example, to have the exponential learning window based on the pairing protocol, we need two pulses with appropriate delay times that are repeated at each second. By sweeping their delay times, various time differences will be obtained. These input pulses are connected to the two ports of TSTDP component which were defined above, so, in the architecture of this component, the arrival times of postsynaptic pulse and presynaptic pulse are determined and the weight change is calculated in both minimal and full models. All reported experiments in this paper assume the nearest spike interaction, which considers the interaction of a spike only with its two immediate succeeding and preceding nearest neighbors. The values of the amplitudes of the weight changes are the reported data of [2] that are given in Tables 1 and 2. As reported in [2], these parameters are set in such a way that the normalized mean square error across all experimental protocols is minimal. The additional parameters τ þ ¼ 16:8 ms and τ ¼ 33:7 ms are taken from [7]. A block diagram of our model is shown in Fig. 2. As described above, triplet learning rules can reproduce triplet and quadruplet
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Table 1 List of the parameters used to model the visual cortex data set as reported in [2]. Parameter
A2þ A3þ A2 A3 τ x ðmsÞ τ y ðmsÞ
Model Full
Minimal
8:8 10 11
0
5:3 10 2
5 10 2
6:6 10 3
8 10 3 0
3:1 10 714 40
3
– 40
Table 2 List of the parameters used to model the hippocampal culture data set as reported in [2]. Parameter
A2þ A3þ A2 A3 τ x ðmsÞ τ y ðmsÞ
Model Full
Minimal
5:3 10 3
4:6 10 3
9:1 10 3
9:1 10 3
3 10
3
7:5 10 9 575 47
3 10 3 0 – 48
experiments. They can also reproduce frequency effects, so we tested our model with these effects too. The required experimental protocols are pairing protocol, triplet protocol and quadruplet protocol. In the pair-based STDP protocol, 60 pairs of pre and post synaptic spikes with a delay of Δt ¼ t post t pre and repetition frequency of 1 Hz are transmitted [1,7]. For triplet experiments, two patterns of spikes that consist of 60 triplets of spikes with repetition frequency of 1 Hz are used. One of these patterns is composed of one post synaptic and two pre synaptic spikes in a pre-post-pre configuration and the other is composed of two post synaptic and one pre synaptic spikes in a post-pre-post configuration. In the first one, the timing differences are defined as Δt1 ¼ t post t pre1 and Δt2 ¼ t post t pre2 , where t pre1 and t pre2 denote the first and second presynaptic spikes of the triplet. In the second one, they are defined as Δt ¼ t post1 t pre and Δt2 ¼ t post2 t pre , where t post1 and t post2 are the first and second postsynaptic spikes of the triplet respectively. Also, quadruplet protocol consists of 60 quadruplets of spikes repeated at a given frequency of 1 Hz. This protocol is composed of either a post–pre pair with a delay of Δt1 ¼ t post1 t pre1 o 0, or pre–post pair with a delay of Δt2 ¼ t post2 t pre2 4 0. In this protocol, the time T is defined by T ¼ ððt post2 þ t pre2 Þ=2Þ ððt post1 þ t prt1 Þ=2Þ, when the post-pre pair precedes the pre-post pair, T is positive and when the opposite happens, T is negative. Identical to Ref. [2], in all quadruplet experiments in this paper, we took Δt ¼ Δt1 ¼ Δt2 ¼ 5 ms.
4. Simulation results The results of TSTDP simulation with VHDL-AMS in Ansoft Simplorer are given in Figs. 3 to 6. The simulation results of [2] are also inserted into these figures to show the close fit of our model to what was reported in this original triplet article that had an optimization scale with the highest analogy to the experimental data. To test our model on a consistent set of data, we took the measurements of [6,7] as are reported in Tables 3 and 4. These available experimental data are inserted to these figures. Fig. 3 shows that the proposed
Fig. 2. The block diagram of our model.
model can reproduce the exponential learning window produced by both minimal and full TSTDP models under the conventional pairing protocol. As it is shown, when the postsynaptic spike precedes the presynaptic one (when Δt ¼ t post t pre is more than zero) the potentiation of the synapse occurs, i.e., Δw 4 0. The depression of the synapse or Δw o 0 is occurred when Δt ¼ t post t pre is less than zero. By increasing the absolute values of Δt, the absolute values of Δw decreased. The vertical bars shown in this figure are the available experimental data reported in Table 4 for pairing protocol which are at Δt ¼ 10 ms and Δt ¼ 10 ms. Fig. 4 depicts that this model succeeds to reproduce the triplet experiments with its minimal and full models. In this figure the values of our results and the experimental data, which are reported in Table 4, in pre-post-pre and post-pre-post combinations of spikes for four different values of Δt1 and Δt2 are shown. In Fig. 5 the simulation results of the minimal and full models for the quadruplet experiments are illustrated. This figure shows the weight change of the synapse as a function of T for Δt ¼ 5 ms. For both T40 and To0, the value of Δw decreases when T is increased. Similar to Fig. 3, in this figure the experimental data of the quadruplet
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Fig. 3. Pairing protocol produced by the proposed model. (a) The weight change in minimal model, (b) The weight change in full model. The required parameters are taken from Table 2.
Fig. 4. Triplet protocol produced by the proposed model.(a) the weight change in minimal model of post-pre-post combination of spikes, (b) the weight change in full model of post-pre-post combination of spikes, (c) the weight change in minimal model of pre-post-pre combination of spikes, (d) the weight change in full model of pre-post-pre combination of spikes. The required parameters are taken from Table 2.
Fig. 5. Quadruplet protocol produced by the proposed model. (a) The weight change in minimal model,(b) the weight change in full model. The required parameters are taken from Table 2.
Fig. 6. Weight change in pairing protocol as a function of pairing frequency ρ reproduced by the proposed model.(a) the weight change in minimal model, (b) the weight change in full model. The required parameters are taken from Table 1.
protocol for 3 values of T( 88.5 ms, 83.7 ms, and 20 ms), as indicated in Table 4, are inserted as vertical bars. Finally, as Fig. 6 shows, these proposed minimal and full models have a BCM like behavior for the
positive and negative values ofΔt. The experimental data of the BCM rule, according to what is indicated in Table 3 (Δt ¼ 10 ms and Δt ¼ 10 ms ), are inserted into this figure to show their good
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Table 3 Experimental weight change as a function of the delay and the rate of spikes in pairing protocol as reported in [6]. ρðHzÞ
Δt ¼ 10 ms Δw
Δt ¼ 10 ms Δw
0.1 10 20 40 50
0.04 70.05 0.14 70.1 0.29 70.14 0.53 70.11 0.56 70.26
0.29 7 0.08 0.417 0.11 0.34 7 0.1 0.56 7 0.32 0.75 7 ;0.19
Table 4 Experimental weight change as a function of the delays in pairing, quadruplet and triplet protocols as reported in [7]. Pairing
Δt 10 ms 10 ms
Δw 0:25 7 0:05 0:177 0:05
Quadruplet Δt ¼ 5 ms
TðmsÞ 88:5 83:7 20
Δw 0:003 7 0:03 0:06 7 0:04 0:21 7 0:04
Triplet (2-pre,1-post)
Δt1 5 10 15 5
Δt2 5 10 5 15
Δw 0:017 0:04 0:03 7 0:04 0:01 7 0:03 0:21 7 0:04
Triplet (1-pre,2-post)
Δt1 5 10 15 5
Δt2 5 10 5 15
Δw 0:33 7 0:04 0:34 7 0:04 0:22 7 0:08 0:29 7 0:05
matching. For validation of the proposed model, some different patterns of spikes including spike pairs, spike triplets and spike quadruplets were used. As it is obvious from these figures, the proposed model has similar results to the mentioned experimental observations. It can be seen that all the results are approximately the same as the results reported in [2]. In order to test the proposed model under the mentioned protocols, full TSTDP and minimal TSTDP rules are used, and it has been demonstrated that the minimal model can reproduce very similar results to those of the full TSTDP model. 5. Conclusion This paper presented a new modeling of TSTDP using VHDL-AMS with Ansoft Simulator. As behavioral simulations show the major characteristics of the nonlinear system such as biological systems, design errors are detected early and predicted. The goal of behavioral modeling in higher level building blocks is to ease and accelerate the design of analog and mixed circuits, reducing simulation time without linear approximations, being implemented. The proposed model is a synapse that can be employed in various neuromorphic systems that are implemented in analog or digital circuits, aiming for different applications and having physical realization of them. This helps the designers to use an efficient methodology to realize physical implementation in neuromorphic VLSI circuits. References [1] G. Bi., M. Poo, Synaptic modifications in cultured hippocampal neurons: dependence on spike timing, synaptic strength, and postsynaptic cell type, J. Neurosci. 18 (24) (1998) 10464–10472.
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Afsaneh Shahimaeen was born in Shiraz, Iran, in 1983. She received the B.Sc. degree in electronics engineering from Razi University, Kermanshah, Iran, in 2006, where she is currently working toward the M.Sc degree in Electronic Engineering at Razi University, Kermanshah. Her current research interests include analog circuits design, artificial intelligence and neural networks.
Gholamreza Karimi was born in Kermanshah, Iran in 1977. He received the B.S. and M.S. and PhD degrees in electrical engineering from Iran University of Science and Technology (IUST) in 1999, 2001 and 2006 respectively. He is currently an Assistant Professor in Electrical Department at Razi University, Kermanshah, since 2007. His research interests include low power analog and digital IC design, RF IC design, modeling and simulation of RF mixed signal IC and microwave devices and artificial intelligence systems.