Bilinearity in Spatiotemporal Integration of Synaptic Inputs
For information processing , a neuron receives and integrates thousands of synaptic inputs from its dendrites and then induces the change of its membrane potential at the soma ., This process is usually known as dendritic integration 1–3 ., The dendritic integration of synaptic inputs is crucial for neuronal computation 2–4 ., For example , the integration of excitatory and inhibitory inputs has been found to enhance motion detection 5 , regularize spiking patterns 6 , and achieve optimal information coding 7 in many sensory systems ., They have also been suggested to be able to fine tune information processing within the brain , such as the modulation of frequency 8 and the improvement of the robustness 9 of gamma oscillations ., In order to understand how information is processed in neuronal networks in the brain , it is important to understand the computational rules that govern the dendritic integration of synaptic inputs ., Dendritic integration has been brought into focus with active experimental investigations ( see reviews 1 , 10 and references therein ) ., There have also been many theoretical developments based on physiologically realistic neuron models 11 , 12 ., Among those works , only a few investigate quantitative dendritic integration rules for a pair of excitatory and inhibitory inputs 3 , 13 and there has yet to be an extensive investigation of the integration of a pair of excitatory inputs or a pair of inhibitory inputs ., In this work , we propose a precise quantitative rule to characterize the dendritic integration for all types of synaptic inputs and validate this rule via realistic neuron modeling and electrophysiological experiments ., We first develop a theoretical approach to quantitatively characterize the spatiotemporal dendritic integration ., Initially , we introduce an idealized two-compartment passive cable model to understand the mathematical structure of the dendritic integration rule ., We then verify the rule by taking into account the complicated dendritic geometry and active ion channels ., For time-dependent synaptic conductance inputs , we develop an asymptotic approach to analytically solve the cable model ., In this approach , the membrane potential is represented by an asymptotic expansion with respect to the input strengths ., Consequently , a hierarchy of cable-type equations with different orders can be derived from the cable model ., These equations can be analytically solved order by order using the Greens function method ., The asymptotic solution to the second order approximation is shown to be in excellent agreement with the numerical solutions of the original cable model with physiologically realistic parameters ., Based on our asymptotic approach , we obtain a new theoretical result , namely , a nonlinear spatiotemporal dendritic integration rule for a pair of synaptic inputs: the summed somatic potential ( SSP ) can be well approximated by the summation of the two postsynaptic potentials and elicited separately , plus an additional third nonlinear term proportional to their product , i . e . , ( 1 ) The proportionality coefficient encodes the spatiotemporal information of the input signals , including the input locations and the input arrival times ., In addition , we demonstrate that the coefficient is nearly independent of the input strengths ., Because the correction term to the linear summation of and takes a bilinear form , we will refer to the rule ( 1 ) as the bilinear spatiotemporal dendritic integration rule ., In the remainder of the article , unless otherwise specified , all the membrane potentials will be referred to those measured at the soma ., We note that our bilinear integration rule is consistent with recent experimental observations 3 ., In the experiments 3 , the rule was examined at the time when the excitatory postsynaptic potential ( EPSP ) measured at the soma reaches its peak for a pair of excitatory and inhibitory inputs elicited concurrently ., We demonstrate that our bilinear integration rule is more general than that in Ref ., 3:, ( i ) our rule holds for a pair of excitatory and inhibitory inputs that can arrive at different times;, ( ii ) our rule is also valid at any time and is not limited to the peak time of the EPSP;, ( iii ) our rule is general for all types of paired synaptic input integration , including excitatory-inhibitory , excitatory-excitatory and inhibitory-inhibitory inputs ., Our bilinear integration rule is derived from the two-compartment passive cable model ., We then validate the rule in a biologically realistic pyramidal neuron model with active ion channels embedded ., The simulation results from the realistic model are consistent with the rule derived from the passive cable model ., We further validate the rule in electrophysiological experiments in rat hippocampal CA1 pyramidal neurons ., All of our results suggest that the form of the bilinear integration rule is preserved in the presence of active dendrites ., As mentioned previously , there are thousands of synaptic inputs received by a neuron in the brain ., We therefore further apply our analysis to describe the dendritic integration of multiple synaptic inputs ., We demonstrate that the spatiotemporal dendritic integration of all synaptic inputs can be decomposed into the sum of all possible pairwise dendritic integration , and each pair obeys the bilinear integration rule ( 1 ) , i . e . , ( 2 ) where denotes the SSP , denotes the individual EPSP , denotes the individual inhibitory postsynaptic potential ( IPSP ) , , , and are the corresponding proportionality coefficients with superscripts denoting the index of the synaptic inputs ., We then confirm the bilinear integration rule ( 2 ) numerically using realistic neuron modeling ., The decomposition of multiple inputs integration in rule ( 2 ) leads to a graph representation of the dendritic integration ., Each node in the graph corresponds to a synaptic input location , and each edge connecting two nodes represents the bilinear term for a pair of synaptic inputs given at the corresponding locations ., This graph evolves with time , and is all-to-all connected when stimuli are given at all synaptic sites simultaneously ., However , based on simulation results and experimental observations , we can estimate that there are only a small number of activated synaptic integration , or edges in the graph , within a short time interval ., Therefore , the graph representing the dendritic integration can indeed be functionally sparse ., Finally , we comment that , in general , it is theoretically challenging to analytically describe the dynamical response of a neuron with dendritic structures under time-dependent synaptic conductance inputs ., One simple approach to circumvent this difficulty is to analyze the steady state of neuronal input-output relationships by assuming that both the synaptic conductance and the membrane potential are constant 3 , 12 ., Such analyses can be applied to study dendritic integration , but they usually oversimplify the description of the spatial integration , and fail to describe the temporal integration ., Another approach to circumvent the difficulty is to study the cable model 14 , 15 analytically or numerically ., For the subthreshold regime , in which voltage-gated channels are weakly activated , the dendrites can be considered as a passive cable ., Along the cable , the membrane potential is linearly dependent on injected current input ., This linearity enables one to use the Greens function method to analytically obtain the membrane potential with externally injected current ., In contrast , the membrane potential depends nonlinearly on the synaptic conductance input 12 ., This nonlinearity greatly complicates mathematical analyses ., Therefore , in order to solve the cable model analytically , one usually makes the approximation of constant synaptic conductance 16 , 17 ., The approximation can help investigate some aspects of dendritic integration , however , the approximation in such a case is not sufficiently realistic because the synaptic conductances in vivo are generally time-dependent ., On the other hand , one can study the dendritic integration in the cable model numerically ., The compartmental modeling approach 14 enables one to solve the cable model with time-dependent synaptic inputs ., This approach has been used to investigate many aspects of dendritic integration ., For instance , it was discovered computationally that dendritic integration of excitatory inputs obeys a certain qualitative rule , i . e . , EPSPs are first integrated nonlinearly at individual branches before summed linearly at the soma 18 , 19 , which was verified later in experiments 20 , 21 ., Clearly , the computational approach can help gain insights into various phenomena of spatiotemporal dynamics observed at the dendrites , however , a deep , comprehensive understanding often requires analytical approaches ., Note that this point has also been emphasized in Ref ., 22 ., Here , our analytical asymptotic method can solve the cable model with time-dependent synaptic inputs analytically and reveal a precise quantitative spatiotemporal dendritic integration rule , as will be further illustrated below ., We begin to study the spatiotemporal dendritic integration of a pair of excitatory and inhibitory inputs ., An analytical derivation of the bilinear integration rule is described in the section of Derivation of the Rule ., The details of the cable model used in the derivation can be found in the section of Materials and Methods ., The validation of the bilinear integration rule using the realistic neuron modeling and electrophysiological experiments is described in the section of Validation of the Rule ., The spatial dependence of the coefficient in the rule is described in the section of Spatial Dependence of ., So far we have addressed the dendritic integration for a pair of excitatory and inhibitory inputs ., A natural question arises: how does a neuron integrate a pair of time-dependent synaptic conductance inputs with identical type ?, The dendritic integration of excitatory inputs has been extensively investigated in experiments ( reviewed in Ref . 1 ) , yet a precise quantitative characterization is still lacking ., According to our idealized cable model , given a pair of excitatory inputs with input strengths and at locations and and at times and , the dynamics of the membrane potential on the dendrite is governed by the following equation: ( 22 ) with the initial and boundary conditions the same as given in Equations ( 4 ) – ( 6 ) ., Similarly , we can represent its solution as an asymptotic series and solve it order by order to obtain the following bilinear integration rule: ( 23 ) where and are EPSPs induced by two individual excitatory inputs , and is the SSP when the two excitatory inputs are present ., Similar to the case of a pair of excitatory and inhibitory inputs , the shunting coefficient only depends on the excitatory input locations and the input time difference ., It does not depend on the EPSPs amplitudes ., Here will still be referred to as a shunting coefficient because the origin of the nonlinear integration for the paired excitatory inputs is exactly the same as that for the paired excitatory and inhibitory inputs from the passive cable model ., The bilinear integration rule ( 23 ) is found to be consistent with the numerical results obtained using the same realistic pyramidal neuron model as the one used in the section of Bilinear Rule for E–I Integration ., For a pair of excitatory inputs with their locations fixed on the dendritic trunk , the rule holds when the amplitude of each EPSP is less than ., For the case of concurrent inputs , at the time when one of the EPSPs reaches its peak valueis found to be linearly dependent of , as shown in Fig . 6A ., This linear relationship indicates is independent of the amplitudes of the two EPSPs ., In addition , as shown in Fig . 6B , the bilinear integration rule is numerically verified in the time interval , for , within which the amplitude of EPSPs are relatively large ., For the case of nonconcurrent inputs , the bilinear integration rule is also numerically verified in the same way , as shown in Fig . 6C–D ., In addition , we find that when the input strengths become sufficiently strong so as to make the depolarized membrane potential too large , i . e . , there is a deviation from the bilinear integration rule ( 23 ) ., This deviation can be ascribed to the voltage-gated ionic channel activities in our realistic pyramidal neuron model ., After blocking the active channels , the rule becomes valid with a different value of for large EPSPs amplitudes , as shown in Fig . 7 ., However , we note that , regardless of input strengths , the amplitude of SC is always two orders of magnitude smaller than the amplitude of SSP ., Therefore , the integration of two excitatory inputs can be naturally approximated by the linear summation of two individual EPSPs , i . e . ., We then perform electrophysiological experiments with a pair of excitatory synaptic inputs to confirm the linear summation ., As expected , this linear summation is also observed in our experiments for both concurrent and nonconcurrent input cases , as shown in Fig . 6E and 6F , respectively ., Note that , the linear summation is also consistent with experimental observations as reported in Ref ., 24 ., Similarly , for a pair of inhibitory inputs , we can arrive at the following bilinear integration rule from the cable model: ( 24 ) where and are IPSPs induced by two individual inhibitory inputs , and is the SSP when the two inhibitory inputs are present ., Here , is the shunting coefficient that is independent of the IPSPs amplitudes but is dependent on the input time difference and input locations ., The above bilinear integration rule ( 24 ) is consistent with our numerical results using the realistic pyramidal neuron model , as shown in Fig . 8A–D ., Our electrophysiological experimental observations further confirm this rule , as shown in Fig . 8E–H ., In the previous sections , we have discussed the integration of a pair of synaptic inputs ., In vivo , a neuron receives thousands of excitatory and inhibitory inputs from dendrites 2 ., Therefore , we now address the question of whether the integration rule derived for a pair of synaptic inputs can be generalized to the case of multiple inputs ., Our theoretical analysis shows that , for multiple inputs , the SSP can be approximated by the linear sum of all individual EPSPs and IPSPs , plus the bilinear interactions between all the paired inputs with shunting coefficients , , and respectively ( the superscript labels the synaptic inputs ) , i . e . , ( 25 ) We next validate the rule ( 25 ) using the realistic pyramidal neuron model ., It has been reported that , for a CA1 neuron , inhibitory inputs are locally concentrated on the proximal dendrites while excitatory inputs are broadly distributed on the entire dendrites 25 ., Based on this observation , we randomly choose 15 excitatory input locations and 5 inhibitory input locations on the model neurons dendrites ( Fig . 9A ) ., In the simulation , all inputs are elicited starting randomly from to ., In order to compare Equation ( 25 ) with the SSP simulated in the realistic neuron model , we first measure , , and pair by pair for all possible pairs ., We then record all membrane potential traces and induced by the corresponding individual synaptic inputs ., Our results show that the SSP measured from our simulation is indeed given by the bilinear integration rule ( 25 ) , as shown in Fig . 9B and 9C ., In contrast , the SSP in our numerical simulation deviates significantly from the linear summation of all individual EPSPs and IPSPs ., According to our bilinear integration rule ( 25 ) , the dendritic integration of multiple synaptic inputs can be decomposed into the summation of all possible pairwise dendritic integration ., Therefore , we can map dendritic computation in a dendritic tree onto a graph ., Each dendritic site corresponds to a node in the graph and the corresponding shunting component is mapped to the weight of the edge connecting the two nodes ., We refer to such a graph as a dendritic graph ., The dendritic graph is an all-to-all connected graph if all stimuli are given concurrently ( Fig . 10A ) ., However , the dendritic integration for all possible pairs of synaptic inputs is usually not activated concurrently in realistic situations ., For instance , if the arrival time difference between two inputs is sufficiently large , there is no interaction between them ., The activated level of the nonlinear dendritic integration for a pair of synaptic inputs can be quantified by the SC amplitude—the weight of the edge in the graph ., The simulation result shows that the number of activated edges at any time is relatively small on the dendritic graph ( Fig . 10B–D ) , compared with the total number of edges on the all-to-all connected graph ( Fig . 10A ) ., Therefore , for the case of a hippocampal pyramidal neuron , the dendritic graph could be functionally sparse in time ., The functional sparsity of a dendritic graph may also exist in neocortical pyramidal neurons ., In vivo , a cortical pyramidal neuron receives about synaptic inputs 26 ., Most of them are from other cortical neurons 27 , 28 , which typically fire about 10 spikes per second in awake animals 29 , 30 ., Thus , the neuron can be expected to receive synaptic inputs per second ., The average number of synaptic inputs within ( membrane potential time constants in vivo ) is ., The number of activated dendritic integration pairs within the interval is , which is relatively small compared with the total possible synaptic integration pairs ., Therefore , the activated integrations or edges in the dendritic graph within a short time window can be indeed functionally sparse ( ) ., In general , the neuronal firing rates vary across different cell types , cortical regions , brain states and so on ., Therefore , based on the above estimate , in an average sense , the graph of dendritic integration is functionally sparse ., Our bilinear dendritic integration rule ( 21 ) is consistent with the rule previously reported 3 , but is more general in the following aspects:, ( i ) Our dendritic integration rule holds at any time and is not limited to the time when the EPSP reaches its peak value ., ( ii ) The rule holds when the two inputs are even nonconcurrent ., This situation often occurs because the excitatory and inhibitory inputs may not always arrive at precisely the same time ., ( iii ) The form of the rule can be extended to describe the integration between a pair of excitatory inputs , a pair of inhibitory inputs , and even multiple inputs of mixed-types ., The spatiotemporal information of synaptic inputs interaction is coded in the shunting coefficient , which is a function of the input locations and input arrival time difference ., Our bilinear integration rule holds in the subthreshold regime for a large range of membrane potential ., When we derive the bilinear rule from the passive cable model , we assume that the input strengths or the amplitudes of membrane potentials require to be small ., This assumption forms the basis of the asymptotic analysis , because the second order asymptotic solutions of EPSP , IPSP and SSP converge to their exact solutions as the asymptotic parameters and ( denoting the excitatory and inhibitory input strengths ) approach zero ., In general , in the passive cable model , the bilinear rule will be more accurate for small amplitudes of EPSPs and IPSPs than large amplitudes ., Importantly , the assumption holds naturally that in the physiological regime when EPSP amplitude is less than 6mV and IPSP amplitude is less than -3mV , and are small ., However , even for EPSP amplitude close to the threshold , i . e . , 10mV , which is unusually large physiologically , we can show that the second order asymptotic solution can still well approximate the EPSP with a relative error less than 5% ., Thus the bilinear rule is still valid for large depolarizations near the threshold ., The validity of the bilinear rule for large membrane potentials is also confirmed in both simulations and experiments ., In particular , in the analysis of our experimental data , to validate the bilinear rule , we have already included all the data when the EPSP amplitude is below and close to the threshold because we have only excluded those data corresponding to the case when a neuron fires ., Our bilinear dendritic integration rule ( 21 ) is derived from the passive cable model ., However , the simulation results and the experimental observations demonstrate that the form of dendritic integration is preserved for active dendrites ., Additional simulation results show that for the same input locations , the shunting coefficients are generally larger on the active dendrites than those on the passive dendrites with all active channels blocked ., We also note that the value of in simulation is different from the value measured in experiments ., This difference may arise from the fact that some parameters of the passive membrane properties , such as the membrane leak conductance , may not be exactly the same as those in the biological neuron , and we have only used a limited set of ion channels in simulation compared with those in the biological neuron ., In addition , the input locations in the simulation and the experiments are different , which may also contribute to this derivation ., However , the bilinear form is a universal feature in both simulation and experiment ., By fixing excitatory input location while varying inhibitory input location , our model exhibits that there exists a region in the distal dendritic trunk within which the shunting inhibition can be more powerful , i . e , a larger , than in proximal dendrites ., This result is consistent with what is reported in Ref ., 31 ., Compared with Ref ., 31 , our work provides a different perspective of dendritic computation ., In their work , the multiple inhibitory inputs can induce a global shunting effect on the dendrites ., However , if we focus on the shunting effect only at the soma instead of the dendrites , our theory shows that all the interactions among multiple inputs can then be decomposed into pairwise interactions , as described by the bilinear integration rule ( 25 ) ., In addition , in this work , we focus on the somatic membrane potential that is directly related to the generation of an action potential ., However , it is also important to investigate the local integration of membrane potentials measured at a dendritic site instead of that measured at the soma ., Asymptotic analysis of the cable model can show that our bilinear integration rule is still valid for the description of the integration on the dendrites ., On the dendrites , the broadly distributed dendritic spines with high neck resistances 32 , 33 will filter a postsynaptic potential to a few millivolts on a branch 34 , 35 ., Within this regime our bilinear integration rule is valid ., Note that our rule may fail to capture the supralinear integration of synaptic inputs measured on the dendrites during the generation of a dendritic spike 36 ., However , if the integration is measured at the soma , our rule remains valid even when there is a dendritic spike induced by a strong excitatory input and an inhibitory synaptic input on different branches 3 ., The bilinear integration rule ( 25 ) can help improve the computational efficiency in a simulation of neuronal network with dendritic structures ., By our results , once the shunting coefficients for all pairs of input locations are measured , we can predict the neuronal response at the soma by the bilinear integration rule ( 25 ) ., By taking advantage of this , one can establish library-based algorithms to simulate the membrane potential dynamics of a biologically realistic neuron ., An example of a library-based algorithm can be found in Ref ., 37 ., To be specific , based on the full simulation of a realistic neuron model , we can measure the time-dependent shunting coefficient as a function of the arrival time difference and input locations for all possible pairs of synaptic inputs and record them in a library in advance ., For a particular simulation task , given the specific synaptic inputs on the dendrites , we can then search the library for the corresponding shunting coefficients to compute the neuronal response according to the bilinear integration rule ( 25 ) directly ., In such a computational framework , one can avoid directly solving partial differential equations that govern the spatiotemporal dynamics of dendrites and greatly reduces the computational cost for large-scale simulations of networks of neurons incorporating dendritic integration ., The animal-use protocol was approved by the Animal Management Committee of the State Key Laboratory of Cognitive Neuroscience & Learning , Beijing Normal University ( Reference NO . : IACUC-NKLCNL2013-10 ) ., We consider an idealized passive neuron whose isotropic spherical soma is attached to an unbranched cylindric dendrite with finite length and diameter ., Each small segment in the neuron can be viewed as an RC circuit with a constant capacitance and leak conductance density 11 , 38 ., The current conservation within a segment on the dendrite leads to ( 26 ) where is the membrane potential with respect to the resting potential on the dendrite , is the membrane capacitance per unit area , and is the leak conductance per unit area ., Here , is the synaptic current given by: ( 27 ) where and are excitatory and inhibitory synaptic conductance per unit area and and are their reversal potentials , respectively ., When excitatory inputs are elicited at dendritic sites and inhibitory inputs are elicited at dendritic sites , we have ( 28 ) where ., For a synaptic input of type , is the input strength of the input at the location , is the arrival time of the input at the location , is the input location ., The unitary conductance is often modeled as ( 29 ) with the peak value normalized to unity by the normalization factor , and with and as rise and decay time constants , respectively 38 ., Here is a Heaviside function ., The axial current can be derived based on the Ohms law , ( 30 ) where is the axial resistivity ., Taking the limit , Equation ( 26 ) becomes our unbranched dendritic cable model , ( 31 ) In particular , for a pair of excitatory and inhibitory inputs with strength and received at and , and at time and , respectively , we have ( 32 ) Similarly , for a pair of excitatory or inhibitory inputs with strengths and received at and , and at time and ( ) , respectively , we have ( 33 ) For the boundary condition of the cable model Equation ( 31 )  , we assume one end of the dendrite is sealed: ( 34 ) For the other end connecting to the soma , which can also be modeled as an RC circuit , by the law of current conservation , we have ( 35 ) where is the somatic membrane area , and is the somatic membrane potential ., The dendritic current flowing to the soma , , takes the form of Equation ( 30 ) at ., Because the membrane potential is continuous at the connection point ( 36 ) we arrive at the other boundary condition at : ( 37 ) For a resting neuron , the initial condition is simply set as ( 38 ) In the absence of synaptic inputs , Equation ( 31 ) is a linear system ., Using a impulse input , its Greens function can be obtained from ( 39 ) with the following boundary conditions and initial condition , For simplicity , letting , , , the solution of Equation ( 39 ) can be obtained from the following system , ( 40 ) with rescaled boundary and initial conditions , where ., Taking the Laplace transform of Equation ( 40 ) , we obtain ( 41 ) Combining the two boundary conditions ( is thus eliminated ) , we have ( 42 ) where ( 43 ) whose denominator is denoted as for later discussions ., For the inverse Laplace transform , we need to deal with singular points that are given by the roots of ., It can be easily verified that these singularities are simple poles and is analytic at infinity ., Then can be written as ( 44 ) where is a constant coefficient in the complex domain , and are the singular points ., Then taking the inverse Laplace transform of Equation ( 44 ) , we obtain ( 45 ) Now we only need to solve and in Equation ( 45 ) to obtain the Greens function of Equation ( 40 ) ., We solve the singular points first ., Defining yields ( 46 ) whose roots can be determined numerically ., There are solutions for with for and Next , to determine the factors we use the residue theorem for integrals ., For a contour that winds in the counter-clockwise direction around the pole and that does not include any other singular points , the integral of on this contour is given by ( 47 ) Using Equations ( 42–44 ) and ( 47 ) , we obtain ( 48 ) where ( 49 ) for ., The solution of the original Greens function for Equation ( 39 ) can now be expressed as ( 50 ) We first consider the case when a pair of excitatory and inhibitory inputs are received by a neuron ., Similar results can be obtained for a pair of excitatory inputs and a pair of inhibitory inputs ., For the physiological regime ( the amplitude of an EPSP being less than and the amplitude of an IPSP being less than ) , the corresponding required input strengths and are relatively small ., Therefore , given an excitatory input at location and time , and an inhibitory input at location and time , we represent as an asymptotic series in the powers of and , ( 51 ) Substituting Equation ( 51 ) into the cable equation ( 31 ) , order by order , we obtain a set of differential equations ., For the zeroth-order , we have ( 52 ) Using the boundary and initial conditions Equations ( 34 ) , ( 37 ) , and ( 38 )  , the solution is simply ( 53 ) For the first order of excitation , we have ( 54 ) With the help of Greens function , the solution can be expressed as ( 55 ) here ‘’ denotes convolution in time ., For the second order of excitation , we have ( 56 ) Because is given by Equation ( 55 ) , the solution of Equation ( 56 ) is ( 57 ) Similarly , we can have the first and second order inhibitory solutions , ( 58 ) ( 59 ) For the order of , we have ( 60 ) whose solution is obtained as follows , ( 61 ) For the numerical simulation of the two-compartment passive cable model Equation ( 3 )  , the Crank-Nicolson method 39 was used with time step and space step ., Parameters in our simulation are within the physiological regime 3 , 12 with , , , , , , , ., , , , and ., The time constants here were chosen to be consistent with the conductance inputs in the experiment 3 ., The realistic pyramidal model is the same as that in Ref ., 3 ., The morphology of the reconstructed pyramidal neuron includes 200 compartments and was obtained from the Duke-Southampton Archive of neuronal morphology 40 ., The passive cable properties and the density and distribution of active conductances in the model neruon were based on published experimental data obtained from hippocampal and cortical pyramidal neurons 18 , 19 , 34 , 41–50 ., We used the NEURON software Version 7 . 3 51 to simulate the model with time step ., The experimental measurements of summation of EPSPs or IPSPs in single hippocampal CA1 pyramidal cells in the acute brain slice followed a method described in Ref ., 3 , with some modifications ., A brief description of modified experimental procedure is as follows ., Acute hippocampal slices ( thick ) were prepared from Sprague Dawley rats ( postnatal day 14–16 ) , using a vibratome ( VT1200 , Leica ) ., The slices were incubated at 34°C for 30 min before transferring to a recording chamber perfused with the aCSF solution ( 2ml/min; 30–32°C ) ., The aCSF contained ( in mM ) 125 NaCl , 3 KCl , 2 CaCl2 , 2 MgSO4 , 1 . 25 NaH2PO4 , 1 . 3 sodium ascorbate , 0 . 6 sodium pyruvate , 26 NaHCO3
Neurons process information via integration of synaptic inputs from dendrites ., Many experimental results demonstrate dendritic integration could be highly nonlinear , yet few theoretical analyses have been performed to obtain a precise quantitative characterization analytically ., Based on asymptotic analysis of a two-compartment passive cable model , given a pair of time-dependent synaptic conductance inputs , we derive a bilinear spatiotemporal dendritic integration rule ., The summed somatic potential can be well approximated by the linear summation of the two postsynaptic potentials elicited separately , plus a third additional bilinear term proportional to their product with a proportionality coefficient ., The rule is valid for a pair of synaptic inputs of all types , including excitation-inhibition , excitation-excitation , and inhibition-inhibition ., In addition , the rule is valid during the whole dendritic integration process for a pair of synaptic inputs with arbitrary input time differences and input locations ., The coefficient is demonstrated to be nearly independent of the input strengths but is dependent on input times and input locations ., This rule is then verified through simulation of a realistic pyramidal neuron model and in electrophysiological experiments of rat hippocampal CA1 neurons ., The rule is further generalized to describe the spatiotemporal dendritic integration of multiple excitatory and inhibitory synaptic inputs ., The integration of multiple inputs can be decomposed into the sum of all possible pairwise integration , where each paired integration obeys the bilinear rule ., This decomposition leads to a graph representation of dendritic integration , which can be viewed as functionally sparse .
A neuron , as a fundamental unit of brain computation , exhibits extraordinary computational power in processing input signals from neighboring neurons ., It usually integrates thousands of synaptic inputs from its dendrites to achieve information processing ., This process is known as dendritic integration ., To elucidate information coding , it is important to investigate quantitative spatiotemporal dendritic integration rules ., However , there has yet to be extensive experimental investigations to quantitatively describe dendritic integration ., Meanwhile , most theoretical neuron models considering time-dependent synaptic inputs are difficult to solve analytically , thus impossible to be used to quantify dendritic integration ., In this work , we develop a mathematical method to analytically solve a two-compartment neuron model with time-dependent synaptic inputs ., Using these solutions , we derive a quantitative rule to capture the dendritic integration of all types , including excitation-inhibition , excitation-excitation , inhibition-inhibition , and multiple excitatory and inhibitory inputs ., We then validate our dendritic integration rule through both realistic neuron modeling and electrophysiological experiments ., We conclude that the general spatiotemporal dendritic integration structure can be well characterized by our dendritic integration rule ., We finally demonstrate that the rule leads to a graph representation of dendritic integration that exhibits functionally sparse properties .
computational neuroscience, neuroscience, biology and life sciences, computational biology