Evaluation of the computational efficacy in GPU-accelerated simulations of spiking neurons

Abstract

To understand the mechanism of information processing by a biological neural network, computer simulation of a large-scale spiking neural network is an important method. However, because of a high computation cost of the simulation of a large-scale spiking neural network, the simulation requires high performance computing implemented by a supercomputer or a computer cluster. Recently, hardware for parallel computing such as a multi-core CPU and a graphics card with a graphics processing unit (GPU) is built in a gaming computer and a workstation. Thus, parallel computing using this hardware is becoming widespread, allowing us to obtain powerful computing power for simulation of a large-scale spiking neural network. However, it is not clear how much increased performance the parallel computing method using a new GPU yields in the simulation of a large-scale spiking neural network. In this study, we compared computation time between the computing methods with CPUs and GPUs in a simulation of neuronal models. We developed computer programs of neuronal simulations for the computing systems that consist of a gaming graphics card with new architecture (the NVIDIA GTX 1080) and an accelerator board using a GPU (the NVIDIA Tesla K20C). Our results show that the computing systems can perform a simulation of a large number of neurons faster than CPU-based systems. Furthermore, we investigated the accuracy of a simulation using single precision floating point. We show that the simulation results of single precision were accurate enough compared with those of double precision, but chaotic neuronal response calculated by a GPU using single precision is prominently different from that calculated by a CPU using double precision. Furthermore, the difference in chaotic dynamics appeared even if we used double precision of a GPU. In conclusion, the GPU-based computing system exhibits a higher computing performance than the CPU-based system, even if the GPU system includes data transfer from a graphics card to host memory.

Appendix Hodgkin–Huxley model

The Hodgkin–Huxley model consists of four differential equations. One equation denotes the dynamics of the membrane potential:

$$\begin{aligned} C \frac{dV}{dt} = I - g_{\mathrm {Na}} m^3 h (u-E_{\mathrm {Na}}) - g_{\mathrm {K}} n^4 (V-E_{\mathrm {K}}) - g_L (V - E_L), \end{aligned}$$

(13)

where C is the membrane capacitance, I is an applied current, $g_{\mathrm {Na}}, g_{\mathrm {K}}$, and $g_L$ are respectively the conductances of $\hbox {Na}^+, \hbox {K}^+$, and leak channels, and $E_{\mathrm {Na}}, E_{\mathrm {Na}}$, and $E_L$ are respectively the reversal potentials of $\hbox {Na}^+, \hbox {K}^+$, and leak channels. The model has terms derived from a sodium current, a potassium current, and a leak current. m, n,  and h are gating variables denoted by

$$\begin{aligned} \frac{dm}{dt}= & {} \alpha _m(V) (1 - m) - \beta _m (V) m, \end{aligned}$$

(14)

$$\begin{aligned} \frac{dn}{dt}= & {} \alpha _n(V) (1 - n) - \beta _n (V) n, \end{aligned}$$

(15)

$$\begin{aligned} \frac{dh}{dt}= & {} \alpha _h(V) (1 - h) - \beta _h (V) h. \end{aligned}$$

(16)

$\alpha (V)$ and $\beta (V)$ are voltage-dependent rates:

$$\begin{aligned} \alpha _m(V)= & {} -\, 0.1\times (v+40)/(\exp (-(v+40)/10)-\,1), \end{aligned}$$

(17)

$$\begin{aligned} \beta _m(V)= & {} 4\exp (-\, (v+65)/18), \end{aligned}$$

(18)

$$\begin{aligned} \alpha _n(V)= & {} -\,0.01\times (v+55)/(\exp (-(v+55)/10)-\,1), \end{aligned}$$

(19)

$$\begin{aligned} \beta _n(V)= & {} 0.125\exp (-(v+65)/80), \end{aligned}$$

(20)

$$\begin{aligned} \alpha _h(V)= & {} 0.07\times \exp (-(v+65)/20), \end{aligned}$$

(21)

$$\begin{aligned} \beta _h(V)= & {} 1/(\exp (-(v+35)/10)+1). \end{aligned}$$

(22)

We set the parameters: $I(t) \,{=}\, 20 \, \hbox {mA}, V_{\mathrm {K}} \,{=}\, -\,95 \,\hbox {mV}, V_{\mathrm {Na}} \,{=}\, 57 \,\hbox {mV}, V_L = -\, 50 \,\hbox {mV}, g_{\mathrm {K}} = 36 \,\hbox {mS}/\hbox {cm}^2, g_{\mathrm {Na}} = 120 \,\hbox {mS}/\hbox {cm}^2, g_L \,{=}\, 0.3 \,\hbox {mS}/\hbox {cm}^2, V_{\mathrm {K}} = -\,95 \,\hbox {mV}, V_\mathrm = 57 \,\hbox {mV}, V_L = -\, 50\, \hbox {mV},\, \hbox {and}\, C = 1 \,\upmu \hbox {F}/\hbox {cm}^2$.