Lumping Izhikevich neurons
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Abstract
We present the construction of a planar vector field that yields the firing rate of a bursting Izhikevich neuron can be read out, while leaving the subthreshold behavior intact. This planar vector field is used to derive lumped formulations of two complex heterogeneous networks of bursting Izhikevich neurons. In both cases, the lumped model is compared with the spiking network. There is excellent agreement in terms of duration and number of action potentials within the bursts, but there is a slight mismatch of the burst frequency. The lumped model accurately accounts for both intrinsic bursting and post inhibitory rebound potentials in the neuron model, features which are absent in prevalent neural mass models.
Keywords
Neural mass Izhikevich neuron Bursting Subthreshold behaviorBackground
The Izhikevich neuron is a computationally simple, yet biologically realistic neuron model [1, 2], which belongs to the class of nonlinear integrateandfire neurons. Bifurcation analysis of a general class of nonlinear integrateandfire neurons is given in [3, 4], whereas a more detailed analysis of the quadratic form is given in [5].
When coupled in a complex network, knowledge of the individual neurons does not suffice to predict the network behavior. As large networks are not suitable for bifurcation analysis, one seeks to formulate low dimensional reductions that capture the spacetimeaveraged behavior of the network. An example is the WilsonCowan reduction [6, 7] where the mean population activities are governed by a low dimensional system of ordinary differential equations (ODEs) or integrodifferential equations respectively. Just as many other lumped formulations of neural activity, these models pivot on the relation between the individual cells’ firing rates and aggregate network activity [8, 9, 10].
with x representing either the mean membrane potential or the postsynaptic conductances/currents.
Q is a linear differential operator which characterizes the synaptic dynamics. Here, f depicts the firing rate as function of the population’s own activity x, due to connections within the population, and some external input I. This basic framework is easily extended to include multiple populations or other features, such as spike frequency adaptation and spatial structures (e.g. neural fields). Yet, it should be noted that models of this form only describe the dynamic evolution of populations averaged quantities.
Population density methods, on the other hand, describe the evolution of the distribution of neurons over the state space, which depends on the quantities that are used in the underlying neuron model [11, 12, 13, 14]. In [14] application of the moment closure method is investigated, whose steadystate solution, in its simplest form (closure at the first moment), is the socalled meanfield firingrate model. Next, in [15] this method is applied to arrive at a planar system of switching ODEs for alltoall coupled networks of Izhikevich neurons with different topologies. This system is subsequently analyzed by means of numerical bifurcation methods [16] and direct simulation. Its variables are the mean adaptation and the synaptic input (here the adjective ‘mean’ can be omitted due to the alltoall coupling). One assumption is that the mean adaptation follows the mean firing rate, rather than the individual firing rates. Furthermore a separation of time scales is assumed: the adaptation time scale is assumed to be much larger than the membrane timeconstant.
The analogy with [15] is that we also consider networks of Izhikevich neurons. The novelty is the construction of the Izhikevich Planar Vector Field (IPVF) from which the firing rate function can be read out. The resulting reduced vector field is three dimensional in contrast to the twodimensional vector field obtained by [15]; we keep the mean membrane potential as a state variables. This state variable follows naturally from the construction from the associated planar vector field. We claim that it is necessary to retain the membrane potential, particularly to deal with postinhibitory rebound spikes/ bursts.
This paper is structured as follows: Izhikevich’ spiking neuron model is studied with parameters corresponding to an intrinsically bursting neuron. Using phase plane analysis, the discontinuous resets are approximated with a smooth branch, which relates to the firing rate of the cell. Most importantly, the proposed approximation leaves all of subthreshold dynamics intact. The resulting model is shown to match quantitatively with the original spiking model, with respect to both spike count and spike timing. Upon making several assumptions relating to network connectivity and synchrony, a random network of Izhikevich neurons can be collapsed into a system of ODEs. Finally, we give two examples of random networks, not necessarily alltoall connected, of heterogeneous Izhikevich neurons where we compare the true network behavior with the properties of the corresponding reduced vector field.
Methods
Izhikevich neuron in the phase plane
Indeed, with four parameters the model is able to reproduce many of the key features observed in real neural tissue, e.g. intrinsic spiking/bursting, spikefrequency adaptation, and rebound spikes/bursts. Due to a reset process, present in all other integrateandfire models, the model does not require a fast recovery variable relating to a resonating current. Hence, the variable u is often seen as a slow variable within the neuron, e.g. Calcium concentration.
The main objective is to replace the fast spiking dynamics by a slowly changing variable which represents the neurons firing rate, whilst leaving the subthreshold dynamics intact. In particular:

The right branch of the Vnullcline corresponds with the firing threshold.

The left branch of the Vnullcline characterizes the fast dynamics in the subthreshold regime.

The unullcline left of the firing threshold determines the slow dynamics in the subthreshold regime.

Point B corresponds to the entrance of the bursting regime.

Point C corresponds to the exit of the bursting regime.

Point A corresponds to the undershoot after burst termination.
An interpretation of these criteria, expressed as a set of ODEs, is as follows.
Reduction
Inside the bursting regime the instantaneous firing rate is now readily determined as the reciprocal of the inter spike times.
with ω = 20.
Although a closed expression can be found for this integral, this approach is not pursued here. The primary reason is that the precise position and shape of the additional branch are merely insignificant. Instead, a much simpler expression is chosen which does not obscure the resulting equations as much as the expression for $\stackrel{\u0304}{v}$ would.
where ${n}_{v}^{\prime}$ is the derivative of n_{ v }. It is undesirable to choose low values of κ since the additional branch will be very steep and the stiffness of the dynamical system increases unnecessarily. Large values, on the contrary, will result in a branch which is too gradual, such that both the timing and the subthreshold behavior are affected. Values of κ in the interval [ 0.5,2] give merely identical results; here is chosen for κ=0.8.
Remark
Since the appearance f_{min} in the reduction is unexpected, it is worthwhile to discuss both the necessity and interpretation for this parameter. In the reduced vector field, an equilibrium on the right branch of the V_{ r }nullcline corresponds to a tonically firing neuron, while absence of such an equilibrium corresponds to an intrinsically bursting neuron. In the case of the bursting neuron, passage from the right branch of the V_{ r }nullcline to the left is guaruanteed by raising the u_{ r }nullcline such that possible equilibria are removed. This raise is achieved by the introduction of f_{min}.
Physiologically, f_{min} represents the minimum the firing rate for a neuron at which it can still be considered ‘bursting’. While this depends on the type of cell at hand, it is generally expected that large cells will have lower firing rates (also within bursts) than small cells.
Single cell comparison
for the Izhikevich and the reduced model respectively.
A comparison for the generation of postinhibotory rebound potentials of a single cell in given in Appendix A: Single cell comparison with postinhibitory rebound potentials.
A population of Izhikevich neurons
With the reduction of a single neuron into place, the next natural step is to consider the global activity patterns of a network of neurons. A single population of merely identical cells is considered first, which can be generalized to multiple populations afterwards.
with V_{ i }the membrane potential of neuron i.
In order to deduce the average network activity, a number of assumptions has to be made. Since all of these assumptions are common practice in neural mass modeling (c.f. [17]), we only state them briefly in the following part. The reader, however, is encouraged to challenge these assumptions and reflect on their implications.
 Temporal averaging. First the individual spike trains are averaged by representing them with the firing rate. That is, every neuron is described by the reduced system (6) as described in the previous section. The corresponding synaptic variable satisfies:${\mathit{\text{Qs}}}_{i}=f({V}_{i},{u}_{i},{I}_{\text{ext}}{I}_{\text{syn},i})$
 Randomly connected network. Let W be a random variable representing the synaptic weight of connections, such that all synaptic weights w_{ ij }are assumed to be independently identically distributed as W. Note that if P (W = 0) > 0, the network is not necessarily alltoall connected. Define$\stackrel{\u0304}{w}:=\mathbb{EW},\phantom{\rule{2em}{0ex}}S\left(t\right):=\sum _{i=1}^{N}{s}_{i}\left(t\right)$
 Synchrony. Neurons receiving a similar input S are assumed to be reside closely together in phase space. Hence, the averages$V\left(t\right):=\frac{1}{N}\sum _{i=1}^{N}{V}_{i}\left(t\right),\phantom{\rule{2em}{0ex}}u\left(t\right):=\frac{1}{N}\sum _{i=1}^{N}{u}_{i}\left(t\right)$(10)
 suffice to approximate, c.f. (6)$\begin{array}{ll}\stackrel{\u0307}{V}& =\frac{1}{N}\sum _{i=1}^{N}g({V}_{i},{u}_{i},{I}_{\text{ext}}{I}_{\text{syn},i})\approx g(V,u,{I}_{\text{ext}}{I}_{\text{syn}})\phantom{\rule{2em}{0ex}}\end{array}$(11a)$\begin{array}{ll}\stackrel{\u0307}{u}& =\frac{1}{N}\sum _{i=1}^{N}h({V}_{i},{u}_{i},{I}_{\text{ext}}{I}_{\text{syn},i})\approx h(V,u,{I}_{\text{ext}}{I}_{\text{syn}})\phantom{\rule{2em}{0ex}}\end{array}$(11b)
The reduced model (11) now represents the lumped network activity.
Extensions to multiple populations are easily incorporated in a similar fashion. An inhibitory population, for instance, can be included by introducing another differential operator, say Q_{2}, which corresponds with a slower postsynaptic current, combined with an appropriate reversal potential.
Results
Comparison
Now that the general derivation of a neural mass model based on Izhikevich neurons is discussed, the correspondence with the network of spiking neurons they represent is studied next. Therefore, simulations of both the neural mass and the detailed model are performed and the relation between them is investigated.
First, only the dynamics of a single population of merely identical excitatory neurons are considered, since we are primarily interested in the quantitative and qualitative comparison of both formulations. A model with multiple populations is studied afterwards.
Setup
The random generation of connections between cells will lead to slight inhomogeneities in the network. Additional inhomogeneities are introduced by randomizing the parameters of the individual cells. This type of perturbation will ensure that the synchronization patterns we observe are due to the network interactions, rather than the intrinsic properties of the neurons. Within the reduced model, the expected value of each of these parameters is used.
Simulations
Simulations of the spiking neuron network are performed with Norns — Neural Network Studio^{a}: a dedicated C++ simulation tool with an intuitive Matlab interface. Networks are generated with their connections as described above. The spike trains obtained from Norns are postprocessed within Matlab, which we discuss in detail below.
The formulated neural mass model can, depending on the parameters, become a stiff system of ODEs. In that case, Matlab’s ode23s is used to for numerical integration, while ode45 is used otherwise.
Comparison
To assess the similarity between the models, it is desirable to compare the same physical quantities within both models. In this light, the synaptic activation S appears to be the best choice: it can be determined from individual spike trains and it is the key variable in the lumped model.
For the lumped model, S is readily available from the simulations. In the network of spiking neurons, we reconstruct the synaptic activation of all neurons by convolving their spike trains with the shape of a postsynaptic response.
One excitatory population
This is explained as follows: as all connections in the network are excitatory, the Vnullcline is effectively lifted during the network bursts due to the external input from synapses. When the network burst terminates, the Vnullcline drops to its initial position. This creates the larger range of u which must be traversed for both the spiking and quiescent phase of the bursting, resulting in more spikes and longer interburst intervals respectively.
These network bursts are, apart from a small difference in interburstfrequency, accurately reproduced by the network reduction (11). Although the interburstfrequency is off, the last network burst (starting at t=900 ms) reveals that the amplitude, shape, and duration are each mimicked in great detail by the neural mass model.
Network with rebound bursts
One of the key features of the reduction is that the subthreshold dynamics of the Izhikevich neuron are left intact, and, as a consequence, it is able to generate postinhibitory rebound potentials, c.f. Appendix A: Single cell comparison with postinhibitory rebound potentials. To illustrate this distinct feature of the model, simulations are performed with a network whose rhythms relies heavily on rebound activity.
A simulation of this particular network is shown in Figure 5B by means of raster plots. The pattern arises because a population bursts in E_{2} gives rise to activation of I, which, in turn, causes a synchronized postinhibitory burst in E_{1}. This burst in E_{1} is large enough to initiate the next population burst in E_{2} before intrinsic mechanisms would. This completes the cycle. Figure 5C shows the summed synaptic variables of each population in blue.
The population reduction from this model follows readily from the single population discussed before; for clarity though, the model and its parameters are stated in Appendix B: Details of three population network. The time evolution of the resulting reduced model is shown in red in Figure 5C. Apart from a change in amplitude, the frequency of the reductio60n matches closely with the original spiking neuron network. The fact that the period of this solution matches better than in the single population example, see previous section, could be explained by the following: In the single population model, c.f. Figure 4, the network bursts are initiated when the slow variable u reaches a certain threshold value and are, therefore, dependent on the value of the slow variable at burst termination. Since the network of Izhikevich neurons increases the slow variable u with discrete steps d, a discrepancy is to be expected with respect to the reduced model, which increases u in a timecontinuous manner. Although the slow variable plays a role in the rebound bursts in the network at hand, the timing of the (rebound)bursts depends more so on the synaptic time constants — which are identical in both models.
Next, we note that, apart from a difference in amplitudes, the shapes of the bursts in both populations I and E_{2} are accurately mimicked. The discrepancy is largest in population E_{1}, with respect to both amplitude and burst duration. It appears that it is this gain in E_{1} in amplitude which gives rise to an increased excitatory input onto E_{2}, resulting in a burst with increased magnitude. Subsequently, the stronger burst in E_{2} explains the amplitude gain of the inhibitory population I.
The natural question to ask now is why the postinhibitory rebound bursts in E_{1} are poorly reproduced by the reduced model. The fact that some neurons are quiescent during the bursts, as follows from the raster plot in Figure 5, suggests that inhomogeneities play a role in this; either due to variations in single cell parameters, or to differences in number of network connections. Indeed, further investigation has revealed that, at the level of a single cell, the generation of rebound bursts is particularly sensitive to variations in parameter b. Simulations (not shown) of the spiking neuron network in which for all cells b is set to a constant — rather than drawn from a normal distribution, c.f. Appendix B: Details of three population network — show that the bursts of neurons in E_{1} are more alike than in Figure 5. As a consequence, the result lies closer to the response of the reduced model (in which all inhomogeneities of the original network are averaged); not only for population E_{1}, also for I and E_{2}.
Without the rebound potentials of E_{1}, neurons in E_{2} receive less excitatory input and the frequency of network bursts is merely determined by the intrinsic dynamics of population E_{2}. that the neurons in E_{2} are not subject to bursts, it remains unclear why the difference in amplitude between the network and corresponding reduction is still in place. A partial explanation is that the excitatory background activity in E_{1}, which is missed by the population reduction, lowers the threshold of neurons, such that network bursts are generated faster than without input. With a lower interburst time, the slow recovery variable u has not decayed as much as it would have if the bursts were further apart. As a result of this, the number of spikes within the burst is expected to be lower. Although this effect plays a role, it is expected that other, yet unidentified mechanisms, are also involved.
Discussion
The work presented here shows how the global activity of networks of intrinsically bursting Izhikevich neurons can be described with a lumped model. This lumped model relies, as the majority of reduced models, on the firing rate of individual neurons. Assisted by phase plane analysis of a single neuron an additional term is proposed, which abolishes the need for discontinuous resets. Effectively, the new term divides the phase space in two regions, corresponding with bursting and nonbursting. Within the bursting regime, the neuron’s firing can, once a separation of timescales is assumed, be determined analytically, while in the nonbursting regime, the neuron’s original subthreshold behavior is kept intact. We note that the resulting model is Lipschitz smooth and its phase portrait shares key features with the FitzhughNagumo [18, 19] and MorrisLecar [20] reductions. In our case, however, ‘active periods’ correspond to an entire burst of action potentials rather than a single pulse. Using simulations, the proposed single cell reduction is shown to match well with the original Izhikevich model. Although a small mismatch is present between the models’ bursting frequency, both duration and number of action potentials within the burst are accurately represented.
With the single cell model into place, the scene is set for lumping networks of bursting neurons. The cells’ planar dynamics are augmented with another state variable, which corresponds to the activation of synapses within the population. By employment of assumptions common in neural mass modeling, a reduced network model is formulated, which consists of three state variables representing the population averages of membrane potential, slow current, and synaptic activation. Similar as for the single cell model, simulations are performed to assess the quality of the newly proposed model. For one population of excitatory intrinsically bursting neurons, the reduced model matches closely with the network dynamics. Indeed, the network bursts, which last longer than the bursts of isolated cells, are mimicked by the lumped model in terms of amplitude, duration, and shape. While the model’s burst frequency is lower than the corresponding spiking neuron network, both models’ bursting frequencies in the network are notably lower than the isolated cells.
Here it is important to point out that a very similar result for a single population has been obtained in [15]. The fact that their formulation is expressed in only two state variables, suggests that the third state variable in our formulation is superfluous. We believe, however, that this third state variable, which corresponds to the mean membrane potential, is essential for keeping the neurons’ subthreshold behavior intact. To demonstrate this particular trait of our reduction, a network is simulated whose rhythm depends critically on the subthreshold behavior (c.f. Figure 5). More precisely, neurons belonging to one of the three populations are capable of generating postinhibitory rebound bursts, which affect the network’s behavior at a global scope. Apart from a discrepancy in amplitude, simulations show that the reduced model is able to accurately describe the network activity. By raising the inhibitory reversal potential, such that the inhibition becomes shunting, we show that this particular rhythm is, in both the original spiking neuron network and the reduced model, indeed reliant on the presence of postinhibitory rebound bursts (c.f. Figure 6).
The single cell reduction proposed in this article considers the bursting Izhikevich neuron merely as an example. We indeed assumed the reset value c to be higher than the local minimum of the V nullcline (i.e. c > − 62.5) and d>0. Although this particular formulation might not be applicable when the parameters at hand do not correspond with the bursting type, we believe that the procedure here can be generalized for all relevant choices of parameters. Going even further, it appears to us that, besides the Izhikevich and other integrateandfire models, also higher dimensional models can be reduced in a similar manner; that is, leaving the subthreshold dynamics intact by only adjusting the dynamics in the suprathreshold regime. In this regime, one should seek to eliminate the (very) fast dynamics and determine their contribution on the slow variables. Yet, these fast variables can play a prominent role in the subthreshold dynamics (e.g. to generate postinhibitory rebound potentials). Elimination of all fast variables could therefore have an unanticipated (and hence undesirable) effect on the subthreshold behavior. So caution is required when constructing firing rate reductions.
Endnote
^{a} Software freely available via ModelDB entry 154739: http://senselab.med.yale.edu/modeldb/showmodel.asp?model=154739.
Appendix A: Single cell comparison with postinhibitory rebound potentials
Appendix B: Details of three population network
This appendix provides the details of the three population network described in Section ‘Network with rebound bursts’. The spiking neuron is characterized first, followed by the reduced model.
B.1 Spiking neuron network
Population parameters
Param.  I  E _{1}  E _{2}  Description 

N  100  300  300  Number of cells in population 
a  N(0.02,0.001)  N(0.03,0.001)  N(0.01,0.001)  Decay rate of recovery variable 
b  N(0.2,0.01)  N(0.3,0.01)  N(0.2,0.01)  Slope parameter of u nullcline 
c  N(−55,1)  N(−55,1)  N(−55,1)  Reset value after spike 
d  N(2,0.1)  N(1,0.1)  N(2,0.1)  Increment of u after spike 
I _{ext}  N(0,0.2)  −3  N(6,0.2)  Constant applied background current 
Connectivity parameters
Source  

Dest.  I  E _{1}  E _{2} 
I  p=0.1  
$\stackrel{\u0304}{g}=0.02$  
E _{1}  p=0.3  
$\stackrel{\u0304}{g}=0.02$  
E _{2}  p=0.1  p=0.1  
$\stackrel{\u0304}{g}=0.02$  $\stackrel{\u0304}{g}=0.005$ 
Synaptic parameters
Exc.  Inh.  Descriptions  

α  5^{−1}  10^{−1}  Synaptic decay rate 
E  0  −80  Reversal potential 
B.2 Reduction
where p_{ kl }and g_{ kl }for k,l ∈ {i,1,2} are as in Table 2 and we used (12).
Notes
Acknowledgements
The authors like to thank both reviewers for their constructive feedback and suggestions.
Supplementary material
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