Journal of Intelligent & Robotic Systems (2020) 98:615–626 https://doi.org/10.1007/s10846-019-01068-0 FULL/REGULAR PAPER Training the Stochastic Kinetic Model of Neuron for Calculation of an Object’s Position in Space Aleksandra Swietlicka´ 1 · Krzysztof Kolanowski1 · Rafał Kapela1 Received: 19 February 2019 / Accepted: 15 July 2019 / Published online: 31 July 2019 © The Author(s) 2019 Abstract In this paper we focus on the stochastic kinetic extension of the well-known Hodgkin-Huxley model of a biological neuron. We show the gradient descent algorithm for training of the neuron model. In comparison with training of the Hodgkin-Huxley model we use only three weights instead of nine. We show that the trained stochastic kinetic model gives equally good results as the trained Hodgkin-Huxley model, while we gain on more concise mathematical description of the training procedure. The trained stochastic kinetic model of neuron is tested in solving the problem of approximation, where for the approximated function the membrane potential obtained using different models of a biological neuron was chosen. Additionally, we present a simple application, in which the trained models of neuron connected with the outputs of a recurrent neural network form a system, which is used to calculate the Euler angles of an object’s position in space, based on linear and angular acceleration, direction and the magnitude of Earth’s magnetic field. Keywords Hodgkin-Huxley model · Markov kinetic formalism · Gradient descent · AHRS · Attitude and heading reference system 1 Introduction Researchers use models of a biological neuron in a number of applications, very often in the field of automatics Neuron models which precisely describe the processes that and robotics. Usually these models exhibit only the spiking take place on the membrane (in particular the Hodgkin- nature of a neural cell, without taking into account all of the Huxley model and its kinetic extensions) can be expressed ionic processes that take place in the neuron. Such spiking in the form commonly used for artificial neural networks. neuron models are applied in: The biological neuron model consists of a current input and – robot control (cf. [1], where a neural network built with a voltage output which is coupled with the input; there are Izhikevich neuron models was tested on a model of an also parameters that can be treated as weights [5–7]and arm of a humanoid robot iCub), sigmoid functions that determine the activation functions. – a neural PID controller, where a neuron model is In the Hodgkin-Huxley model considered in [5–7]nine expected to adjust parameters of the controller or to parameters are used as weights. In this paper we show that if approximate differential and integral equations [28], the stochastic kinetic model is used, it is possible to obtain – creation of a visual attention map [29], equally good results of training and gain on less complex – general path planning [13, 21, 30]. mathematical description of the training procedure, due to reducing the number of weights to three and replacing most The neuron model commonly used in the above applica- of differential equations with ’add-and-multiply’ rules. The tions is the Izhikevich model, presented among others in gradient descent algorithm is used in the training procedure. [14]. This model is described with only two equations: v = 0.04v2 + 5v + 140 − u + I, Aleksandra Swietlicka´ u = a(bv − u), (1) [email protected] where v is the potential on the neuron’s membrane, u is the membrane recovery variable and I is the input current (stim- 1 Faculty of Computing, Institute of Automatic Control and Robotic, Poznan University of Technology, ul. Piotrowo 3A, ulus), while a and b represent the time scale and sensitivity 60-965 Poznan,´ Poland of the recovery variable, respectively. In our paper, however, 616 J Intell Robot Syst (2020) 98:615–626 we will consider the stochastic kinetic model of biological neu- 2.1 The Hodgkin-Huxley Model of the Biological ron, which besides the spiking nature of neuron, takes into Neuron account ionic processes that take place on the membrane. In this paper we present the gradient descent algorithm The neuron model created by Alan Lloyd Hodgkin and for training the stochastic kinetic model of neuron. The trained Andrew Huxley in 1952 is based on the idea that the model is used to control the process of calculating the Euler potential on the membrane can be modeled with an angles of an object in space (Roll, Pitch, Yaw) based on equivalent circuit and described with the following equation data collected from three sensors: accelerometer, gyroscope [2, 9, 12, 16, 17]: and magnetometer. These sensors are part of the Attitude dV and Reference Heading System (AHRS) and are created in C = I −g m3h (V − V )−g n4 (V − V )−g (V − V ) , (3) dt Na Na K K L L Micro Electro-Mechanical Systems (MEMS) technology. The AHRS enables calculation of the Euler angles, but the where I is the total membrane current density (input of the neu- AHRS algorithm is computationally expensive in view of ron), V is the potential (output of the neuron), C is matrix calculations, hence the idea of using an artificial neural conductance, while gNa,gK and gL are conductivities and network combined with a model of a biological neuron, VNa,VK,VL are the reverse potentials of specific types of ions - which is less complex than commonly used algorithms. sodium, potassium and chloride, respectively. Values of the The same algorithm of gradient descent was already used conductivities and reverse potentials are shown in Table 1. for the training of a model of a dendritic structure of the The membrane is built from channels which consist of biological neuron given with the following equation [12]: small gates, which in turn control the movement of ions a ∂2V ∂V between the interior and exterior of the neuron. In the = C +g m3h (V − V )+g n4 (V − V )+g (V − V ) (2) 2R ∂x2 ∂t Na Na K K L L Hodgkin-Huxley approach it is assumed that gates can be in either of two states, permissive or non-permissive. This means adopting the algorithm of gradient descent Dependence between these states is described with the in the structure that can be treated as a biological neural following equations: network (not just a single neuron). The way of discretization dn and implementation of this kind of dendritic structure of = αn (V )(1 − n) − βn (V ) n, biological network is presented in [26]. The procedure of dt dm the training of the stochastic kinetic model of a biological = αm (V )(1 − m) − βm (V ) m, neural network and application of the trained model in an dt dh image processing task are shown in [23]. = αh (V )(1 − h) − βh (V ) h, (4) The main objective of the research presented in this paper dt is focused on determining an object’s position in space. Our where the first term (αi (V )(1 − i)) represents the transi- research also includes self-testing of robot’s sensors [15]. tion of a particular gate from the non-permissive to permis- We are still expanding these calculations with new ideas. sive state, while the second term (βi (V ) i) – the transition Models of biological neuron and biological neural networks from the permissive to non-permissive state. are of our interest. We are trying to apply them to solve Sigmoid functions α (V ) and β (V ) are of the following our problems and to analyse their capabilities, hence the form: 0.01 · (10 − V ) V presented research. α V = β V = · − n ( ) 10−V n ( ) 0.125 exp The paper is organized as follows: in Section 2 exp − 1 80 10 shows the Hodgkin-Huxley model and its extension to 0.1 · (25 − V ) V the deterministic and stochastic kinetic model. Section 3 α (V ) = β (V ) = 4 · exp − m 25−V m (5) exp − 1 18 presents the gradient descent algorithm for training of the 10 stochastic kinetic model. Section 4 focuses on experimental V 1 α V = · − β V = results, while Section 5 shows applications of a trained h ( ) 0.07 exp h ( ) 30−V 20 exp + 1 model of the biological neuron. Finally, in Section 6, a short 10 summary is provided. Table 1 The parameters of Hodgkin-Huxley model [ ] [ 2] 2 Model of the Biological Neuron iVi mV gi mS/cm Na 115 120 In this section we will provide a short description of the K −12 36 Hodgkin-Huxley neuron model and its deterministic and L 10.6 0.3 stochastic kinetic extensions. J Intell Robot Syst (2020) 98:615–626 617 2.2 Deterministic Kinetic Model of the Biological A detailed description of the kinetic formalism can be Neuron found in [4]. A kinetic extension of the Hodgkin-Huxley model requires that processes which take place on the membrane are 3 Training Algorithm described with Markov kinetic schemes. The dynamics of the potential change on the membrane of the neuron can be It has already been shown [5–7] that it is possible to written in the following form [22]: train the Hodgkin-Huxley neuron model by manipulating its parameters. In [5–7] α (V ) and β (V ) are rewritten in the dV C = I − g [m h ] (V − V ) − g [n ] (V − V ) − g (V − V ) (6) dt Na 3 0 Na K 4 K L L following form: 0.01 · V − V V − V = αn = · βn αn (V ) V −V βn (V ) 0.125 exp In the kinetic model of neuron, sodium and potassium exp αn − 1 80 gates can be in eight and five states, respectively.