Ultrafast population coding and axo-somatic compartmentalization

Chenfei Zhang1,2,3, David Hofmann4,5, Andreas Neef1,2*, Fred Wolf1,2*

1 Department for Nonlinear Dynamics, Max Planck for Dynamics and Self-
Organization, Göttingen, Germany

2 Bernstein Center for Computational Neuroscience, Göttingen, Germany,

3 Graduate School for Neurosciences, Biophysics, and Molecular Biosciences,
University of Göttingen, Göttingen, Germany

4 Department for Physics, Emory University, Atlanta, GA, USA

5 Initiative in Theory and Modeling of Living Systems, Emory University,
Atlanta, GA, USA

These authors contributed equally to this work.

*<EMAIL_ADDRESS>*<EMAIL_ADDRESS>
## Abstract

Cortical neurons in the fluctuation driven regime can realize ultrafast
population encoding. The underlying biophysical mechanisms, however, are not
well understood. Reducing the sharpness of the action potential onset can
impair ultrafast population encoding, but it is not clear whether a sharp
action potential onset is sufficient for ultrafast population encoding. One
hypothesis proposes that the sharp action potential onset is caused by the
electrotonic separation of the site of action potential initiation from the
soma, and that this spatial separation also results in ultrafast population
encoding. Here we examined this hypothesis by studying the linear response
properties of model neurons with a defined initiation site. We find that
placing the initiation site at different axonal positions has only a weak
impact on the linear response function of the model. It fails to generate the
ultrafast response and high bandwidth that is observed in cortical neurons.
Furthermore, the high frequency regime of the linear response function of this
model is insensitive to correlation times of the input current contradicting
empirical evidence. When we increase the voltage sensitivity of sodium
channels at the initiation site, the two empirically observed phenomena can be
recovered. We provide an explanation for the dissociation of sharp action
potential onset and ultrafast response. By investigating varying soma sizes,
we furthermore highlight the effect of neuron morphology on the linear
response. Our results show that a sharp onset of action potentials is not
sufficient for the ultrafast response. In the light of recent reports of
activity-dependent repositioning of the axon initial segment, our study
predicts that a more distal initiation site can lead to an increased sharpness
of the somatic waveform but it does not affect the linear response of a
population of neurons.

## Introduction

Neurons communicate via brief, all-or-none action potentials (APs) that are
initiated close to the cell body, at the axon initial segment (AIS), and
propagate along the axon. Depending on the degree of axonal branching, a
single neuron can contact tens of thousands downstream neurons, and at every
contact point the action potential is converted to a transient conductance
change in the recipient cells. Central neurons receive thousands of such
inputs every second but only a tiny fraction of this incoming information is
encoded in the timing of their action potentials outputs. The precise timing
of an action potential determines which aspects about the input it can encode.
This action potential timing is ultimately decided by the mechanisms affecting
the initiation of the action potential.

The dynamics of AP initiation is controlled by the voltage dependence of all
conductances in the proximal axon that are active in the vicinity of the
threshold voltage of action potential initiation. Indeed a cortical neuron’s
membrane potential is typically close to threshold and so operates in the
fluctuation driven regime [1]. In particular the voltage dependence of the
sodium channels has been identified as the key determinant of the AP onset
dynamics [2], reflected in two different measurables: i) The slope of the
phase space plot ($\dot{V}(t)$ vs $V(t)$) at the AP onset, often termed onset
rapidness. ii) The precision of action potential timing or, in the frequency
domain, the bandwidth of input frequencies that exert an influence over this
timing.
While the exact shape of the action potential waveform at its onset could be
considered irrelevant for neuronal information processing, the spectral
bandwidth of the encoder, that is a population of neurons, turns out to be of
great importance to the function of neuronal networks. It defines an upper
limit to the information transfer between neurons[3, 4]. Incoming action
potentials could potentially convey sub-millisecond timing since postsynaptic
potentials rise within less than a millisecond, thus the transmission at a
chemical synapse can be so reliable that it adds no relevant extra temporal
jitter [5, 6]. However, this information can only be transmitted, if the
outgoing APs were initiated with a sub-millisecond precision as well. The
timing precision is also relevant for phenomena such as spike timing-dependent
plasticity [7], and the spike time coding in sensory [8, 9] and motor systems
[10, 11].

It is known that a the population firing rate of pyramidal neurons can track
changes in their input almost instantaneously. A population of layer 2/3
pyramidal neurons fires uncorrelated, when driven by fluctuating current,
consistent with uncorrelated synaptic input. This is reminiscent of neuronal
populations in-vivo [12]. When a current step of 20 pA amplitude, as small as
a single synaptic input, is added to each neuron’s input simultaneously, the
population firing rate significantly increases in less than a millisecond [13,
14]. High temporal precision of AP initiation thus translates into ultrafast
population encoding which is thought to be important for realizing fast
information processing and decision making in the brain [15]. The underlying
biophysical mechanisms of ultrafast population encoding and temporally precise
firing of action potentials, however, are not well understood.

Thus a key question in regards of the importance of the action potential
waveform, specifically the onset rapidness, is whether and how it is linked to
the encoding bandwidth.
The voltage dependence of the membrane current around the AP threshold,
determines the AP onset rapidness at the initiation site, and the bandwidth of
the dynamic gain: if the voltage dependence steeply increases around the
threshold, onset rapidness and encoding bandwidth are high. However, it is not
clear whether the observation of an AP high onset rapidness is sufficient to
predict a high bandwidth. Such a predictive power would be very helpful. The
AP waveform is comparably easy to measure but in itself not relevant for
network function. Measuring the encoding bandwidth, on the other hand,
requires thousands of spikes. However, the relation between AP onset rapidness
and encoding bandwidth is further complicated, if the AP waveform is observed
at the soma, tens of micrometers away from the axonal initiation site.

The somatic AP waveform of cortical neurons has a high onset rapidness of
around $30\mathrm{ms}^{-1}$. The phase space plot of the somatic voltage rises
very abruptly at the AP onset, producing a ’kink’. While this has been
interpreted to predict an encoding bandwidth of several hundred Hertz [16],
others have suggested the high somatic onset rapidness to result from a low
onset rapidness at the AIS which is distorted by the propagation towards the
soma [17]. However, pyramidal neurons in acute brain slices and in-vivo have
indeed been found to encode with a bandwidth of $200-400\mathrm{Hz}$ [18, 19,
16, 13] or even $1\mathrm{kHz}$ in the case of human cells [20]. Furthermore,
experimental studies found that a reduced axonal sodium current, either by
partial sodium removal or by blocking of sodium channels, reduced the encoding
bandwidth and the onset rapidness of the somatic AP [14].

Theoretical models can be coarsely categorized according to their morphology
and AP generation. The simplest morphology is a point neuron which completely
ignores all spatial structure [21, 2, 16], spatially extended neurons instead
can account for the separation of AP initiation site from the soma, or effects
of dendritic trees [22, 23, 24, 25]. AP generation in turn can be as simple as
a threshold unit that produces an instantaneous spike as the potential
threshold is crossed or be comprised of an AP dynamics. Surprisingly, the most
simple model, a point neuron model with hard threshold AP generation, the
leaky integrate-and-fire (LIF) model is capable of reproducing the encoding
features of cortical neurons [21]. Point neurons [2, 16] and simple multi-
compartment neurons [26, 27] with biophysically motivated membrane currents or
similar voltage dependence neither have a bandwidth far above the firing rate,
nor do they display a pronounced change of dynamic gain when the input
correlation time is changed. Thus the challenge to understand the observed
phenomena is to isolate features that account for the observed dynamic gain in
biophysically more realistic models.

Several hypotheses have been proposed to explain the high encoding bandwidth
and the sudden onset of the somatic AP: i) cooperative gating of ion channels
[16], ii) dendritic load on the soma [24, 25], and iii) electrical decoupling
between soma and AIS [22, 23]. While i) argues for a mechanism directly
affecting the AP dynamics, ii) and iii) propose a neuron morphology based
argument. Specifically iii) propose an elegant approach based on a reduced
multi-compartment model that provides novel insight in the possible origin of
the somatic waveform ’kink’. Unaccounted for by single compartment neuron
models, the AP initiation site of a cortical neuron is located in the AIS and,
hence, separated from the soma [28, 29]. This separation is proposed to cause
the somatic membrane potential ’kink’. While it was suggested that this may
also account for the observed input-output properties, a direct examination of
the encoding ability has not been performed.

To quantify the population encoding ability, the linear response function of
the population firing rate has been adopted for experimental cell physiology.
The linear response function represents the dynamic gain as a function of
frequency. This concept was first proposed by Knight [30] and later
theoretically elaborated in more biologically realistic and detailed neuron
models [31, 21, 3, 2, 32, 33, 34, 27, 26, 35].

In this work, we first reproduce the somatic ’kink’ due to a more distal
positioning of the AIS in a multi-compartment model similar to the one
proposed in [22]. We then determine the dynamic gain in the fluctuation driven
regime injecting colored noise currents and investigate whether shifting the
AIS along the axon does impact i) the bandwidth of the dynamic gain and ii)
the sensitivity of the dynamic gain to the input correlation time. We
furthermore explore the effects of different voltage sensitivities of the
sodium activation as well as the impact of soma size on the linear input-
output relation.

## Materials and Methods

### Model

All simulations were performed with NEURON 7.3 [36] as a module for Python
2.6. The morphology, that is a ball-and-stick model with soma and axon, and
its parameters are the same as in [22] apart from a slight deviation from the
ball-and-stick morphology by modeling the soma as a cylindrical compartment
instead of a sphere (see Fig. 1A)). The soma has a diameter of $d_{S}=50\mu m$
and length $l_{S}=50\mu{}\mathrm{m}$ while the axon is
$d_{A}=1\mu{}\mathrm{m}$ thick and $l_{A}=600\mu{}\mathrm{m}$ long. The
intracellular resistance is $R_{a}=150\Omega\mathrm{cm}$. The specific
membrane capacitance is $c_{m}=0.75\mu{}\mathrm{F/cm}^{2}$ and the specific
membrane resistance (inverse leak conductance) is
$R_{m}=30000\Omega\mathrm{cm}^{2}$ with a leak reversal potential of
$E_{\mathrm{L}}=-75\mathrm{mV}$. For the axon, the passive currents are the
same as for the soma. Sodium channels are modeled as a NEURON point process at
the axon initial segment at distance $x_{\mathrm{Na}}$ from the soma. As in
[22], we only take the activation dynamics of the sodium conductance into
account and omit inactivation. Peak sodium conductance
$g_{\mathrm{Na}}=5.23\cdot 10^{−3}\mathrm{S/cm}^{2}$. Since this simplified
conductance model gives a dynamical system that does not produce a spike we
define a spike threshold at $V=0$ and reset the voltage over the whole cell to
$E_{\mathrm{L}}$ (which is equivalent to the resting potential for about the
whole cell apart from the immediate neighborhood of the sodium current
position) when the threshold is crossed, analogous to the spiking mechanism in
integrate-and-fire neuron models [37].

The neuron is characterized by the following partial differential equations.
The voltage in the soma is

$c_{m}\frac{\partial V_{S}(x,t)}{\partial
t}=\frac{l_{S}}{4R_{a}}\frac{\partial^{2}V_{S}(x,t)}{\partial
x^{2}}-\frac{V_{S}(x,t)-E_{\mathrm{L}}}{R_{m}}+\frac{I(t)}{\pi
d_{S}}\delta\left(x-\frac{d_{S}}{2}\right)$ (1)

with $I$ being the input current. The axonal membrane voltage is described by

$c_{m}\frac{\partial V_{A}(x,t)}{\partial
t}=\frac{l_{A}}{4R_{a}}\frac{\partial^{2}V_{A}(x,t)}{\partial
x^{2}}-\frac{V_{A}(x,t)-E_{\mathrm{L}}}{R_{m}}-\frac{\bar{g}_{\mathrm{Na}}m(V_{A}(x,t))(V_{A}(x,t)-V_{\mathrm{Na}})}{\pi
d_{A}}\delta\left(x-x_{\mathrm{Na}}\right)$ (2)

and

$\tau_{m}\dot{m}=m_{\infty}(V)-m.$

The boundary conditions of the equations (1) and (2) are given as
$V_{S}(l_{S},t)=V_{A}(0,t)$, $d_{S}\frac{\partial V_{S}(x,t)}{\partial
x}|_{x=d_{S}}=d_{A}\frac{\partial V_{A}(x,t)}{\partial x}|_{x=0}$,
$\frac{\partial V_{S}(x,t)}{\partial x}|_{x=0}=0$, and $\frac{\partial
V_{A}(x,t)}{\partial x}|_{x=l_{A}}=0$.

### Simulations

We drove the model by injecting a Gaussian colored noise current $I(t)$ into
the soma as done in experimental studies [19, 18, 38, 13] and detailed by
equation (1). We realize the colored noise by an Ornstein-Uhlenbeck process
[39, 40] with correlation time $\tau$, standard deviation $\sigma$ and mean
current of $\mu$:

$\tau\mathrm{d}I(t)=(\mu-I(t))\mathrm{d}t+\sqrt{\tau}\sigma\mathrm{d}W(t)$

where $W(t)$ denotes a Wiener process with zero mean and variance of one. The
numerical solution to the above model equations were computed by implicit
Euler integration with temporal steps of $\mathrm{\Delta t}=25\mu{}\mathrm{s}$
and spatial step sizes of $\mathrm{\Delta x}=1\mu{}\mathrm{m}$.

Several conditions were maintained for all calculations to make results
comparable. i) we kept the firing rate constant [2] at $5\mathrm{Hz}$ similar
to the experiments by Tchumatchenko et al. [13]; ii) the neuron was driven in
its linear response regime, that is the input current must not be too large
which we achieve by demanding the standard deviation of the somatic membrane
voltage to be about $5\mathrm{mV}$; iii) we assured that the neuron is driven
by current fluctuations rather than the mean current which is realized by
guaranteeing that no spiking occurs with a constant, non-fluctuating input of
the chosen mean current; and iv) we controlled for the coefficient of
variation (CV) to be in the range $0.85\pm 0.05$ to make sure that the firing
activity is not bursting and close to regular pyramidal cell activity.

#### Parameter space exploration.

We explore the dependence of the dynamic gain on several parameters. Those are
the input current correlation time $\tau\in\\{5,50\\}\mathrm{ms}$, the
position of the sodium currents $x_{\mathrm{Na}}=\\{20,40,80\\}\mu\mathrm{m}$,
the voltage sensitivity $k_{a}=\\{0.1,6\\}\mathrm{mV}$ 111The voltage
sensitivity is a parameter of the activation function
$m_{\infty}(V)=1/(1+\exp((V_{1/2}-V)/k_{a})$., and the soma diameter
$d_{S}=\\{10,50\\}\mu\mathrm{m}$.

### Computing the linear response of a neural ensemble

In the linear response regime we can approximate the instantaneous firing rate
of the ensemble of independent neurons by a linear filter acting on the input
current. To analyze the bandwidth we thus need to compute the transfer
function of this filter. While there are different ways to estimate the
transfer function from spiking data and the stimulus current [19, 13], here we
use the approach proposed in [18] as outlined in what follows.The transfer
function is given as

$G\left(f\right)=\frac{|\mathcal{F}\left(C_{I\nu}(\tau)\right)|}{|\mathcal{F}\left(C_{II}(\tau)\right)|}$

where $\mathcal{F}$ denotes the Fourier transform, $C_{I\nu}$ and $C_{II}$ is
the input-output correlation and input auto-correlation function,
respectively. The Fourier transform of an auto-correlation function is the
power spectrum according to the Wiener-Khinchin theorem whose analytical form
is known for an Ornstein-Uhlenbeck process as
$\mathcal{F}(C_{II}(\tau))=P(f)=\frac{2\tau\sigma^{2}}{1+((2\pi f)^{2}}$.
To compute input output correlation function we make use of the following
equality

$C_{I\nu}(\tau)=\langle\delta
I(t-\tau)\delta\nu(t)\rangle=\frac{1}{NT}\int\mathrm{d}t\delta
I(t-\tau)\sum_{i,j}\delta(t-t_{i}^{j})$

$\delta(\cdot)$ is the Dirac delta, $\delta\nu$ and $\delta I$ denote small
deviations. This illustrates that we can conveniently get $C_{I\nu}(\tau)$ by
computing the spike-triggered average (STA). For computing the STA we need to
select a time window large enough for correlations to be decay, here we choose
a window of $800\mathrm{ms}$ centered about the spike time. We then computed
the Fourier transform and applied a filter bank of Gaussians in the complex
plane to de-noise before computing the power of the signal by taking the
absolute value.

We then performed this computation of the dynamic gain for each parameter
value by running a numerical simulation of the stochastic current driven model
described above for 20.5s (with 0.5s of initialization to make sure the
initial conditions of the model don’t affect the results and 20s of actual
recording). Given the 5Hz firing rate we gather on average 100 spikes per run.
We repeat this simulation 50 times and compute a single STA from the resulting
time series.
In order to estimate the regime of statistical accuracy of our simulation we
perform two analyses. First we compute a confidence interval by bootstrapping
and second we compare the result to the null hypothesis that the transfer
function stems from random spike times. Bootstrapping is achieved simply by
choosing 400 samples with replacement from the set of 400 spike triggered
current time series to compute the STA. We do this 1000 times and from those
boot strapped transfer functions we take the upper and lower bound define the
95% confidence interval around the true transfer function. For the null
hypothesis testing we need to shuffle the spike times with the constraint of
keeping the firing rate and ISI distribution. In order to do so we add a
random number to all spike times produced by a single simulation run of 20
seconds. A different number is added to each of the 50 repetitions and the
spike times are computed modulo 20 to assure they don’t exceed the absolute
time window. We then repeat this procedure 500 times and take the upper bound
of the resulting 95% interval as the curve corresponding to the null
hypothesis. We took this curve as delineation of validity of our numerically
computed dynamic gain functions and show only the valid regime above this
significance border.

## Results

We built a two compartment model (Fig 1 A)) with identical characteristics as
proposed in Brette 2013 [22] (see Material and Methods). To make sure that our
model compares to Brette’s in its central features we show the bifurcation of
the voltage at the soma in dependence of the sodium channel’s position. As can
be seen in Fig 1 B) we obtain the same result as shown in Fig. 2 in Brette
(2013) [22]. We further reproduced the decoupling observed in Fig. 1F of [22]
as demonstrated in Fig 1 C). While we found the same loss of voltage control
about -55 mV between the two compartments when clamping and slowly increasing
the somatic voltage, we point out that in the dynamic case (i.e. no clamp is
applied) the decoupling of the compartments is less pronounced. We further
investigated the sharpness of the shape of the somatic phase plot in
dependence of the sodium current position 2A). We found that the speed by
which the voltage rises increases the further distant the sodium channels are
from the soma. However, the key quantity to explore fast population encoding,
the dynamic gain, had not been investigated for this multi compartment model
yet. We thus performed a thorough analysis of the dynamic gain in dependence
of several different model parameters in the following.

Fig 1: Model properties. A) A simple sketch of the model morphology: both soma
and axon are modeled as cylinders. The sodium current is positioned at
$x_{\mathrm{Na}}$ and the stimulus current is injected into the center of the
soma. More simulation and model details are given in the main text. B) We
reproduce the bifurcation property discussed in Fig. 2 of [22]. Sodium current
(red) and lateral current (black) is plotted as function of the voltage at the
axon initial segment for different positions of the AIS (top to bottom:
$x=20\mathrm{\mu m},27\mathrm{\mu m},40\mathrm{\mu m}$). C) Voltage at the
axon initial segment as a function of somatic voltage. Patch clamped (black)
curve in comparison with the dynamic case (red). Analogous to Fig. 1F of [22],
the patch clamped case shows a loss of voltage control around
$-55\;\mathrm{mV}$. However, we point out that the dynamic case shows no
sharp, localized transition. This hints that the loss of voltage control is
less pronounced in the dynamic case, that is during an action potential. D)
Gain is plotted as a function of input current frequency for the sub-threshold
responses to a colored noise stimulus with correlation time
$\tau=5\mathrm{ms}$. We investigate the sub-threshold transfer function for
three different positions of sodium channels, colors denoted in the legend.

#### Somatic kink is not sufficient for a high information bandwidth.

First we asked whether the dynamic gain depends on the position of the sodium
channels. As can be seen in Fig. 2 B) the gain curves do not show a higher
cut-off frequency when the position of sodium channels is moved further away
from the soma. The low bandwidth we observe stands in contrast with previous
experimental findings demonstrating that pyramidal neurons do have a cut-off
frequency in the regime of hundreds of Hz. Instead the cut-off frequency is
the same as previously found in conductance based single compartment models
[2].

#### Input correlations don’t alter bandwidth.

Stimulating the neuron with colored noise current simulates the barrage of
synaptic input to a neuron with the current correlation time reflecting an
effective synaptic time constant. Previous studies have shown that in simple
model neurons as the LIF and other models with a large onset rapidness show
sensitivity to this input correlation time. More recently experimental
evidence has been provided in support of this effect [13]. We find that the
present ball-and-stick multi-compartment conductance based model fails to
reproduce this dependence on input correlations for any of the different
sodium channel positions (Fig. 3).

Fig 2: Phase plot and transfer functions. A) Phase plot for the soma, i.e. we
plot time derivative of the voltage over the voltage. Colors correspond to
different sodium channel positions $x_{\mathrm{Na}}$ (confront legend in B) ).
The vertical straight line results from the artificial resetting mechanism in
this model. B) Comparison of transfer functions for different positions of the
sodium current. The input current correlation time is set to
$\tau=5\mathrm{ms}$. All curves are normalized such as to start with a gain of
one for better comparison of the dynamic gain’s tail. 95% confidence intervals
are plotted as shaded areas, however, for the large part of the transfer
function these are overlapping with the average curve. All curves are above
statistical significance (detailed explanation in Material and Methods).  Fig
3: Effect of input correlation time on dynamic gain. We plot the transfer
functions for different input current correlation times to investigate whether
the model shows dependence of the input correlation time, also called the
Brunel effect [21]. Model properties are the same as for Fig. 2. Graph colors
code different sodium channel positions with the same color code as in Fig. 2,
the distance increases from the left to the right panel as
$x_{\mathrm{Na}}=20,40$ and $80\mu\mathrm{m}$. Thin lines correspond to small
correlation times ($\tau=5\mathrm{ms}$) and thick lines correspond to large
correlation times ($\tau=50\mathrm{ms}$)

#### High voltage sensitivity recovers empirical findings.

We have shown that somatic onset rapidness does not imply fast population
coding as quantified by the dynamic gain. These empirical findings, however,
can be recovered when changing the voltage sensitivity parameter $k_{a}$ from
$6\mathrm{mV}$ to $0.1\mathrm{mV}$, that is, making the sodium current
activation curve steeper and, hence, the neuron model more voltage sensitive
at the AIS. In Fig. 4A) we show that i) the compartment decoupling becomes
more pronounced for all sodium channel positions tested if voltage sensitivity
is increased and ii) We note that these voltage plots are recorded in the
dynamic regime, not by performing a static voltage clamp measurement.
Importantly, Fig. 4B) shows that the dynamic gain decays slower and, hence,
the information bandwidth is increased as had been reported for single
compartment models in prior work [2, 33, 27].
Next we explored the effect of sodium channel position on the dynamic gain for
the model with high voltage sensitivity and found that there is no substantial
difference in terms of information bandwidth between the conditions (Fig. 5).
Furthermore, we compared different input current correlation times and found
that higher correlation times do increase the information bandwidth as had
first been found by Brunel, et al. for the leaky integrate and fire neuron
[21].

Fig 4: High voltage sensitivity. A) For different sodium current positions
(confront legend), we plot the axonal voltage at the position of the sodium
currents over the somatic voltage. Solid lines are voltage traces for low
voltage sensitivity $k_{a}=6$, dashed lines denote the same traces for high
voltage sensitivity $k_{a}=0.1$. B) We compare transfer functions for
different voltage sensitivities.  Fig 5: High voltage sensitivity and sodium
channels position. Dynamic gain curves are plotted for A) a short input
current correlation time of $\tau=5\mathrm{ms}$ and B) a long correlation time
of $\tau=50\mathrm{ms}$. We plot dashed lines to consistently denote linear
response functions for models with high voltage sensitivity. Shaded areas
define the 95% confidence interval. Colors denote different sodium current
positions as explained in the legend.

#### Soma size affects the dynamic gain for the highly voltage sensitive
model.

The last parameter that we tested was the somatic diameter $d_{\mathrm{S}}$.
As $50\mu\mathrm{m}$ is a reasonable upper limit for cortical neurons we
tested $10\mu\mathrm{m}$ which demarcates the lower end of the cortical soma
size spectrum [37]. Since for low voltage sensitivity we could not find any
significant changes when reducing the soma size, in Fig. 6 we show the linear
response functions for varying soma sizes in the case of high voltage
sensitivity. We find that reducing the soma size increases the bandwidth
effectively bringing the transfer functions to resemble more those of a LIF.
Indeed this is not surprising given that we approach the limit where our two
compartment model becomes a single compartment model whose response function
characteristics are well studied. However, our results show that our findings
are indeed robust with respect to soma size and don’t depend on an
artificially large soma as compared to most cortical cells.

Fig 6: Effect of soma size. We investigate the soma size effect on the linear
transfer function for different scenarios (confront plot titles). As in
previous plots, dashed lines denote high voltage sensitivity and shaded areas
the 95% confidence interval.

## Discussion

Understanding the biophysics of neuronal action potentials has a long history
with substantial successes in modeling and experiment [41, 42] and is
fundamental to our understanding of neuronal information processing. An aspect
of current interest are the features of the neuronal action potential and the
morphological and electrical organization of nerve cells that can enable high
bandwidth ultrafast encoding. Interestingly, a substantial fraction of models
fail to exhibit a realistically high cut-off frequency of hundreds of Hz of a
population of neurons [18, 13] as well as the experimentally observed
sensitivity to the input correlation time of the dynamic gain [13] while a
simplistic neuron model like the LIF can account for those features [21].
However, due their abstract nature and simplicity such models do not allow us
to further investigate the biophysical features that are responsible for these
measurable phenomena. Several hypotheses have been put forward as possible
biophysical explanations: i) ion channel cooperativity [16], ii) dendritic
load [24, 25], and iii) compartmentalization of soma and axon through more
distal position of the AIS [22, 23].

In this study we investigated the model proposed in [22], a simplified and
analytically tractable conductance based multi-compartment model that was
proposed to explain the rapid action potential onset [22, 23]. The key feature
of this model is that rapid onset in the somatic membrane potential can be
achieved by moving the position of the AIS further away form the soma and thus
compartmentalizing the neuron, viz. electrically decoupling the soma and axon.

Interestingly, a series of experimental studies have shown that the AIS can
undergo plastic changes due to spiking activity and specifically relocate its
position along the axon [43, 44, 45, 46, 47]. These studies raise the question
as to whether AIS position plays a role in neural coding. We investigated
whether the position of the AIS can impact the linear response of a population
of neurons. We employed the simplified ball-and-stick neuron model that was
proposed in [22] and investigated whether relocation of the AIS can reproduce
the two key experimental findings: high cut-off frequency of the dynamic gain
and input correlation time dependence of the dynamic gain. In doing so we also
shed light on the relationship between rapid onset and large bandwidth.

While we reproduced the somatic rapid AP onset for distal positions of the AIS
as shown in [22] we found that the linear response remains unaffected by the
specific position of the AIS. Furthermore, we found that the linear response
of this model is insensitive to changes in input correlation time. Besides
revealing that this reduced model cannot account for ultrafast neural
responses as found in experiments, our findings also highlight that a rapid
onset at the soma is not sufficient for producing a large encoding bandwidth.
We show that one way to reconcile experimental results with this model is by
tuning the activation slope of the sodium activation function which can be
done by increasing the voltage sensitivity of the sodium channels. While we
get more realistic cut-off frequencies the results, however, are still not
sensitive to the position of the AIS.

We conclude thus that first our study suggest that rapid onset at the action
potential initiation site can be linked to rapid responses of a neuron
ensemble but not somatic rapid onsets since those can be caused by mechanisms
not responsible for the high neural bandwidth. However, we point out that
reliably measuring AIS membrane potentials pose an outstanding experimental
challenge that has yet to be solved [48, 49]. Second, pure dislocation of the
AIS does not affect population coding. This is a prediction that can be tested
experimentally to further elucidate the biophysical factors underlying high
bandwidth ultrafast population encoding. On the one hand, if the prediction is
correct models need not precisely account for this morphological aspect when
studying population dynamics. However, other morphological factors like the
dendritic tree size have been shown to be relevant [24, 25] and thus further
theoretical investigation is needed to identify an adequate minimal model that
can guide the identification of the biophysical properties that determine
dynamic population coding.

## Acknowledgments

This work was partially supported by the Federal Ministry for Education and
Research (BMBF) under grant no. 01GQ1005B (to A.P., T.G., F.W. and D.B.),
through CRC 889 by the Deutsche Forschungsgemeinschaft and by the
VolkswagenStiftung under grant no. ZN2632 (to F.W.), and by the China
Scholarship Council. We thank Rainer Engelken and Barbara Feulner for fruitful
discussions.

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# Training Students’ Abstraction Skills Around a Café 2.0

Géraldine Brieven,Lev Malcev,Benoit Donnet  0000-0003-1410-1470
0000-0002-5259-4336 0000-0002-0651-3398 Université de Liège, Institut
Montefiore, Belgium

###### Abstract

Shaping first year students’ mind to help them master abstraction skills is as
crucial as it is challenging. Although abstraction is a key competence in
problem-solving (in particular in STEM disciplines), students are often found
to rush that process because they find it hard and do not get any direct
outcome out of it. They prefer to invest their efforts directly in a concrete
ground, rather than using abstraction to create a solution.

To overcome that situation, in the context of our CS1 course, we implemented a
tool called Café 2.0. It allows students to actively and regularly practice
(thanks to a longitudinal activity) their abstraction skills through a
graphical programming methodology. Moreover, further than reviewing students’
final implementation, Café 2.0 produces a personalized feedback on how
students modeled their solution, and on how consistent it is with their final
code. This paper describes Café 2.0 in a general setting and also provides a
concrete example in our CS1 course context. This paper also assesses students’
interaction with Café 2.0 through perception and participation data. Finally,
we explain how Café 2.0 could extended in another context than a CS1 course.

###### Index Terms:

Café 2.0, Computer-Assisted Learning, Automatic Feedback, Abstraction,
Graphical Reasoning, CS1, Programming Challenge, Programming Environment

##  I Introduction

For a first year student, entering Higher Education is a completely other
world opening to them compared to Secondary School. First-year students are
expected to digest the topics they are taught on their own, with very limited
guidance from the supervisors. On one hand, students should become autonomous
as it is part of the abilities they should develop and, on the other hand,
supervisors cannot often dedicate enough time to each student individually,
since they belong to a large and heterogenous group. Moreover, the subjects
students have to integrate are often larger and more complex, making the
transition even steeper from Secondary School to Higher Education. This
situation contributes to a high failure rate, as well as a high withdrawal
ratio throughout the year [1], in particular in CS1 classes [2, 3, 4]. This is
especially true in our country, where open access to Higher Education is the
rule (with some exceptions in the Medicine and Engineering Faculties). The
consequence is that we cannot make any kind of assumptions about a first year
student’s background. This can be a significant drawback in some areas, like
Computer Science [5], that strongly rely on Mathematical skills. Shaky
foundations in Mathematics leads to poor abstraction capacities as well as a
lack of rigor in problem solving, while abstraction and problem solving
represent the core of the Computer Science curriculum. However, many students
are not fully aware of that until they take the first evaluation, which
reveals where they stand with respect to the curriculum requirements. In many
cases, that check point already comes too late in the semester : students feel
demotivated and cannot make up the time they have lost as new topics keep
being taught.

Figure 1: Motivation and context around Café 2.0 and our CS1 course.

In that teaching context, one of our goals in our Introduction to Programming
class (usually refered to as “CS1”) is to maintain students onboard. To do so,
regular activities are organised [6] to give them the opportunity to be
actively learning and to receive feedback. Most of their productions mobilize
abstraction skills since any Computer Scientist should demonstrate such
abilities [7, 8]. More generally, abstraction skills are involved in all STEM
(Science, Technology, Engineering, and Mathematics) disciplines because they
support problem and solution modeling, whatever the nature of the problem [9].
Thanks to abstraction, a whole class of equivalent problems (where only input
parameters vary) can be addressed, rather than only some specific instances.
In our course, students apply abstraction through the Graphical Loop Invariant
Based Programming (Glibp) [10], consisting in representating and manipulating
a drawing reflecting a solution supported by an iterative process (i.e., a
loop in a piece of code). That context and goals are illustrated in the
rectangle labeled “Objectives” in Fig. 1.

Encompassing that rectangle, Fig. 1 expresses how abstraction can be taught
with respect to those objectives. In practice, the only way to make a large
group of students active and provide them with a correct and personalized
feedback is to transit through a remote activity supported by an online
system. We implemented such a learning tool called Café 2.0 (standing for
“Correction Automatique et Feedback des Étudiants”). It is currently applied
only in the context of our CS1 class in which students are exposed to C
programming language concepts and a graphical programming methodology (Glibp).
Café 2.0 automatically assesses students’ programming exercises and provides
students with high quality feedback and feedforward information (i.e., what
should they do to improve their solution). One key point of Café 2.0 is that
it does not only focus on the program output but also on the cognitive
abstraction process inherent to the program construction, which differs from a
large majority of learning tools restricting to automatic code simulation and
assessment [11, 12, 13, 14, 15, 16, 17, 18, 19]. That differentiation gives to
Café 2.0 the potential to be transposed in other (STEM) disciplines that would
rely on a sequential resolution process founded on a scheme constructed
upstream, similarly to our course.

To make students use a learning tool, it needs to be integrated into the
course via activities, as shown in the outer rectangle labeled “Activities” in
Fig. 1. In our course, the Programming Challenge Activity (Pca) [20] was
created. It spans the four-month semester by regularly addressing statements
students should solve by submitting some solutions as many times as they want.
Furthermore, for each submission, students receive personalized feedback
(especially detailed feedback about their solution modeling through a drawing)
and feedforward. In addition to that activity, it is aimed to offer students
another kind of opportunity to train abstraction skills. To meet that purpose,
the GameCodes [21] are currently being implemented. Contrary to the Pca, the
GameCodes give students more freedom and guidance across their resolution.

In this paper, we carefully depict Café 2.0 features, by defining them from a
generic and specific perspective. Doing so, we highlight the interest and
potential of Café 2.0 out the scope of our CS1 class. Then, this paper
discusses two remote activities : the Pca (implemented over Café 2.0) and the
GameCodes (under implementation). After that, we report how students receive
the Pca and Café 2.0 during our course organized during Academic Year
2022–2023. Finally, this paper exposes how Café 2.0 could be extended to
support new problem profiles (from other STEM fields). In particular, we
define the checklist a resolution flow should follow to get the best from Café
2.0.

The remainder of this paper is organized as follows: Sec. II depicts Café 2.0,
while Sec. III discusses activities in Café 2.0; Sec. IV presents perception
and participation data we collected; Sec. V explains how Café 2.0 should be
extended to support additional disciplines; Sec. VI positions this paper with
respect to the state of the art; finally, Sec. VII concludes this paper by
summarizing its main achievements.

##  II Café 2.0

Café 2.0 has been implemented as a new version of Café 1.0. Initially, Café
1.0 emerged as a set of Python scripts integrated in a submission plateform
[22]. As shown in Fig. 2, to interact with Café 1.0, the students need to
solve some statement, format their solution in a text file with predefined
placeholders, and then upload it on a submission plateform. Each submission is
instantaneously processed by Café 1.0 that computes the grades, highlights
what should be adapted in the current submission (through the feedback), and
provides pointers to the theoretical courses (through the feedforward). In
this way, students get the opportunity to realize their misunderstanding and
improve their subsequent submissions. Fig. 2 also highlights that, as
supervisors, in addition to be timesaving and scalable, such a system allows
us to keep track of student’s behavior by collecting basic data related to
their activity and performance.

Figure 2: Illustration on how students interact with Café 1.0.

However, in practice, that initial version has several drawbacks, i.e., ($i$)
the absence of interactiveness (since Café 1.0 has no interface), ($ii$) the
lack of guidance across the resolution (since it has to be handled beforehand)
[23], and, ($iii$), the limited learning analytics collected. That prevents a
student from tracking their progress and learning journey. Likely, those
limitations were repeling some students from taking the full benefits from
that learning approach. That is what leveraged the development of Café 2.0,
turning Café 1.0 into a fully online plateform and expanding it in order to
make students’ online-learning journey richer and more natural. The different
upgrades from Café 1.0 are detailed in Sec. II-A.

###  II-A Café 2.0 Overview

In this subsection, Café 2.0 is presented from two perspectives : a generic
one (providing a high level overveiw of Café 2.0, so that the reader can
project it more easily in their own discipline ground) and a specific one
(allowing to embody that generic view).

(a) Generic modules

(b) Modules as implemented in our CS1 course.

Figure 3: High level overview of Café 2.0 infrastructure.

From a generic point of view, Fig. 3a illustrates the different
functionalities Café 2.0 is currently offering. Those functionalities target
two types of end users: the students and the supervisors. Considering the
students, they can interact with three different modules : ($i$) the Activity
Resolution, where students can pick some statement, solve it, and submit it in
order to receive a personalized feedback and feedforward; ($ii$) the Drawing
Editor, equipping students with graphical components they can drag and drop in
order to design some solution according to some outline being shown and
detailed during the course ; ($iii$) the Progress Tracker, whose goal is to
depict where students stand in their learning journey, with respect to their
activity and performance on the tool. Besides this, a supervisor interacts
with three modules that echo the ones intended for the students : ($i$) the
Statement Encoding, through which a supervisor can define a new statement and
parametrize its automatic assessment, feedback, and feedforward, in the
context of an activity ; ($ii$) the Drawing Editor, that may be used to define
a new statement, if the supervisor wants to include some schemes to be
completed ; ($iii$) the Learning Analytics Dashboard, illustrating the
students’ learning behavior, based on the collected learning analytics. Such a
feature will automatically distill students’ performance thanks to the
personalized feedback that is constructed, which also allows us to identify
precisely the points of misunderstood material.

In Fig. 3a, the modules that are usable via an interface (i.e., frontend) are
comprised in the central rectangle (“Interface”). Then, the backend is
represented on the right where the main data that needs to be stored to
support those modules is represented through cylinders. Finally, the backend
also includes the Correction and Feedback functionality responsible for
handling a student’s submission, knowing to which statement it is supposed to
respond to and relying on a misconception library where typical mistakes made
by students have been stored.

Next, from a particular point of view, Fig. 3b refreshes Fig. 3a by specifying
the different modules that have just been exposed in the context of our CS1
course. More precisely, Fig. 3b focuses on the modules that are currently
implemented (colored in Fig. 3a) and numbers the flow that must be followed by
a statement, from its definition to its resolution (that may rely on several
improvement iterations). Considering the different modules, it can be noticed
that the generic Drawing Editor defined previously has been instantiated as
Glide. It provides pre-defined patterns and tutorials, specific to the
programming methodology being taught in our course [10]. Next, the activity
that is proposed as a concrete opportunity to practice is the Programming
Challenge Activity (Pca) [20]. It mainly consists in addressing some
statements to the students whose expected resolution namely relies on an
interactive blank outline (referred by Blank Graphical Loop Invariant) to fill
in, as shown on the right of Fig. 3b. All those modules implemented in the
context of our CS1 course are detailed in the next subsections.

###  II-B Activity Resolution Module

(a) Generic resolution flow through productions.

(b) Resolution flow relying on the Graphical Loop Invariant.

Figure 4: Resolution module of Café 2.0. Figure 5: Blank Graphical Loop
Invariant in the Pca. It also shows how our tool follows the Glibp methodology
with tabs, one for each step of the resolution process.

Similarly to Café 2.0’s overview, the Activity Resolution is presented at two
levels : a generic one and a specific one. First, Fig. 4a shows that any
resolution supported by Café 2.0 should be composed of an abstraction phase
followed by a concrete phase. Then, for each phase, those can include one or
more sequential productions [24], one production being illustrated in a
rounded rectangle. Additionally, some productions may also overlap those two
phases in order to bridge them based on specific configurations of their
solution. Finally, some locks can be defined between the productions, as
represented through the diamond “edited”. In Fig. 4a, it comes just after the
“Main Representation”, meaning that students need to first work on it before
deriving some specific states and developing their solution. In regards to
that, Fig. 4b depicts that resolution process in the context of our CS1
course. It results in five productions in our case. More concretely, those are
also shown through Fig 5. On that figure, the four tabs above support the
abstraction of the solution via the Blank Graphical Loop Invariant (described
in Sec. II-D) and its transposition into concrete states thanks to movable
bars. In particular : ($i$) “GLI” consists in filling the Blank Graphical Loop
Invariant, turning it as students’ own Graphical Loop Invariant; ($ii$)
“Initial Representation” requires students to graphically manipulate their
Graphical Loop Invariant to reflect the initial configuration of their
solution. In this way, they can derive how the variables supporting their
solution should be declared and initialized in their code; ($iii$) “Final
Representation” corresponds to graphical manipulation of the Graphical Loop
Invariant to illustrate the final solution so that students can deduce under
which condition their loop stops; ($iv$) “Loop Variant Function” is for
proposing a function that gives the number of elements that still need to be
processed in order to get the final solution. All those tabs are supposed to
be done in sequence. Students have also access to the code editor (bottom part
of Fig. 5, labelled as “Code Editor”). The code may be pre-filled with a
template [25] students must edit with their code. Students have also access to
the “playground” mode in which they can compile and test their pieces of code.
Once students are ready, they can submit their whole solution.

At that point, two interests in decomposing students resolution into pre-
defined productions can be highlighted. First, it allows Café 2.0 to pave
students’ resolution with respect to a given methodology. Next, it frames
their solution, making feasible an automatic correction and personalized
feedback mainly thanks to the Blank Graphical Loop Invariant.

###  II-C Correction and Feedback Module

(a) Generic feedback module.

(b) Example of feedback based on a given misconception library.

Figure 6: Correction and Feedback module of Café 2.0.

Fig. 6 presents how a given student’s solution gets instantaneously assessed
and commented, based on a predefined misconception library. Like previously,
that process is represented in a general way (through Fig. 6a) while Fig. 6b
instantiates it in the context of our course. Both figures show that the
correction and feedback are performed based on a misconception library
containing typical mistakes students tend to make. It is worth noticing that
the misconception concept is broadly used in the STEM literature[26, 27]. In
this paper, we similarly use the terms “misconception”, “error”, and “mistake”
to point out “something that is done wrong”.

####  II-C1 The misconception library

From a general point of view, any supervisor wanting to use Café 2.0 as an
automatic assessment and feedback system should define a rubric checklist [28]
beforehand, forming so the misconception library. That rubric should be
organised according to the productions a student’s submission is made up of.
For each production, typical mistakes should be identified (based on previous
experiences, like presented in other studies [29]). Then, each mistake should
be characterized by a unique error code, a nature (syntactic/semantic), a
gravity factor (quantifying how serious the mistake is), a feedback message
(explaining in details the error), and, optionally, a corresponding reference
to the course (i.e., feedforward). Once the misconception library has been
fed, some respective rule-based checks must be implemented and simply
configured in order to catch each mistake based on a given submission.

####  II-C2 The Correction and Feedback Construction

When those last set-ups are ready, the system can process a given submission.
For a given student’s submission, each production is digested by a dedicated
checker module that detects any potential mistake defined in the misconception
library. If a mistake is captured, the student’s final grade gets impacted
with respect to the gravity factor characterising the mistake. In addition to
this, the corresponding feedback message and reference to the course are added
in the list of comments being provided to the student, eventually. That list
of comments is splited on a per production basis.

Fig. 6b shows an example of misconception library from which mistakes were
detected in a given submission. Once the feedback has been received, a student
may improve their solution and submit it again.

###  II-D Production Modeling

For a given production (a production being represented in a rounded rectangle
in Fig. 4), in order to be able to capture errors, a tradeoff must be found
between constraining the solution and letting enough freedom to students. The
more bounded the solution, the more predictable the students’ answers with
respect to the typical mistakes, which can be caught through rule-based
checks. Besides this, The more freedom students get, the easier the
transposition of their own reasoning to the provided canvas. In practice, one
way is to shape each production and model the expected solution using blank
solution components whose semantic and relations between each others must be
specified beforehand. Typically, in our course, solution components stand as
fields to fill, instructions in the code, or components of the Blank Graphical
Loop Invariant (being movable bars and boxes). In particular, the Blank
Graphical Loop Invariant is a blank drawing depicting only the general shape
that should follow a correct and rigorous Graphical Loop Invariant [10, 24].
Students must then annotate properly the figure so that the drawing becomes
their Graphical Loop Invariant modeling their own solution. An example of
Blank Graphical Loop Invariant is provided in Fig. 7. Any Blank Graphical Loop
Invariant always comes with two types of boxes: ($i$) red boxes standing to
host expressions (i.e., constants, variables, operations, or left blank) and
are to be completed by students without support; ($ii$) green boxes standing
to host labels that students must drag and drop from a pre-defined list (see
the list on the left of the Blank Graphical Loop Invariant in Fig. 5).

Figure 7: Blank Graphical Loop Invariant as solution ground.

That list contains multiple choices, some of them being the expected answers,
others being purely random. Doing so, we pave the way for an automatic
correction of the Graphical Loop Invariant (with strong feedback and
feedforward). This can be achieved thanks to the fact each box is numbered. In
this way, when a student’s solution gets corrected, each piece of the solution
is easily pointed out, allowing to bring a rich feedback while still keeping
it clear and smooth to digest for the student. To define a Blank Graphical
Loop Invariant, a supervisor can use the drawing editor Glide.

###  II-E Drawing Editor Module

Figure 8: Screenshot of Glide.

The Graphical Loop Invariant Drawing Editor (Glide) proposes to supervisors
all the components a Graphical Loop Invariant can be composed of so that they
can build up some Blank Graphical Loop Invariant, responding to a given
problem.

Besides this, Glide also helps students in drawing their own Graphical Loop
Invariant by proposing pre-defined graphical components [10] students must
arrange and fill in. Fig. 8 shows a final representation of a Graphical Loop
Invariant, illustrating how to compute the product of all integers belonging
to a range specified as input. Furthermore, students can be guided across
their composition by activating some step-by-step tutorials. Once a student
considers their Graphical Loop Invariant is completed, they can submit it and
some basic checks are performed. In particular, syntactic mistakes are
detected (such as the lowerbound being further than the upperbound or some
description of what has been achieved so far that is missing). However, the
Graphical Loop Invariant semantic is not verified, which means that the
solution can be positively assessed by the Glide although the Graphical Loop
Invariant does not make sense.

Similarly to that particular instance, a Drawing Editor could be implemented
in other fields by equipping it with the adequate graphical components,
modeling them in order to define rules on and drawing up a step-by-step
tutorial guiding students in handling those components.

##  III Designing an Activity

Automatic assessment and feedback can be fully beneficial only if it gets
encapsulated in some activities with a sound pedagogical reflection behind
[30]. In this section, two activities are introduced : the Programming
Challenge Activity (Pca) [20] and the GameCodes [21] . The Pca is already
available in Café 2.0 while the GameCodes are currently under construction.

###  III-A The Pca (relying on the Blank Graphical Loop Invariant)

Figure 9: Café 2.0 activity timeline over the semester.

The Pca is made up of six Challenges. A Challenge is a statement aligned to
one or several theoretical chapters taught the week(s) before. In the fashion
of the chapters in our CS1 course, the Challenges are cumulative, requiring a
good level of understanding about the previous topics to be properly handled.
Each Challenge consists in producing some pieces of code. For Challenges 2, 3
and 4, students must also graphically model their solution by filling some
Blank Graphical Loop Invariant [10]. Regarding the modalities, any Challenge
starts on Wednesday, 06:00PM and finishes on Friday, 08:00PM. During this
2-days timeframe, a student can submit up to three times their solution, each
one receiving an automatic feedback and feedforward. The latest submission
determines the final mark and each Challenge accounts for 2% of the final mark
for the course. After that certificative period, students are free to keep
training, but it will not affect their final grades. It is worth noting that
the first Challenge (called Challenge 0) does not account in the final grade.
Its only purpose is to make sure students grasped how to use Café 2.0.
Finally, each student has the opportunity to play a trump card allowing them
to skip one of the Challenges (that, will not count towards the student’s
mark) [20]. Fig. 9 illustrates the timeline of the Pca for Academic Year
2022–2023. The six Challenges were spread between September, 13 and December,
16. The blocus (i.e., a period during which no classes are organized and
students are supposed to prepare themselves for the upcoming exams) is
organized between December, 16 and January 3 for Academic Year 2022–2023. Fig.
9 also shows when the midterm and the final exam were organized.

###  III-B The GameCodes

Contrary to the Pca, the purpose of the GameCodes is to give students the
opportunity to get a personalised learning experience across exercises
resolution [21]. A GameCode corresponds to a large statement to solve through
predefined resolution steps. In our case, one GameCode is defined per chapter
of the course. Students are fully free to take them or not, as they are not
certificative. Furthermore, when they solve a GameCode, students can choose
how to handle it by jumping to the resolution step they want at any time,
asking for tips or theoretical reminders and submitting answers to get
feedback.

##  IV Preliminary Evaluation

This section discusses some observations from the data collected by Café 2.0.
In particular, we focus on how students grasp the tool. Sec. IV-A explains the
data we collected, while Sec. IV-B focuses on metrics of interest.

###  IV-A Dataset

During Academic Year 2022–2023, 97 students ($N_{r}$=97) registered to our CS1
class. Among them, 76% were new comers (first year at University), while 23%
were either repeaters or have changed their programs.

We collected data throughout the semester thanks to Café 2.0 Learning
Analytics module. Among the registered students, a maximum of 80 students
($N_{c}$=80 – 82.5% of $N_{r}$) connected at least once on Café 2.0.

A survey was conducted at the end of the final exam. 74 students ($N_{s}$=74 –
76.3% of $N_{r}$) shared their opinion. The survey included Likert scale and
open questions, all related to different aspects of Café 2.0.

###  IV-B Results

Results include students’ view regarding Café 2.0 (Sec. IV-B1) as well as how
they were active on Café 2.0 (Sec. IV-B2).

####  IV-B1 Perception

This paper addresses a particular focus on three questions students were asked
in the survey. The first question compares the impact of Café 2.0’s experience
on students’ motivation in learning, with respect to other (more classic)
activities. Fig. 10 shows that the Pca (supported by Café 2.0 and described in
Sec. III-A) comes as the second most stimulating activity, after the
theoretical lessons. In particular, 60% of the respondents (strongly) agreed
that the Pca was motivating, 25% had no opinion, and 15% did not embrace that
online experience. More precisely, both the certificative and the formative
periods seem relevant to practice the course. Taking a closer look, the
opinions are slightly stronger regarding the formative period. Likely, on the
one hand, some students were more autonomous, allowing them to take benefit
from the Pca independently from any grading pressure while others did not take
the formative periods as an opportunity to train. This is also underpinned by
Fig. 13b (detailed below). One reason explaining some lower activity outside
the certificative period could be that students got stuck despite the
feedback, preventing them from progressing and achieving the Challenge.

Besides this, a special interest was dedicated to the way students perceive
the automated feedback. From a general point of view, Fig. 11 reflects that
the feedback is well-received by the students. More precisely, half of the
respondents found is clear and understandable, 30% felt mitigated about it,
and 20% could not understand it well. Despite some misunderstanding of the
feedback, a majority of respondents (74%) felt boosted in improving their
solution after receiving the feedback. In the same way, more than 60% of
respondents could identify their gaps, focus on the corresponding theoretical
supports to fill them, and better understand the topic. Lastly, it is also
interesting to note that, although some students were struggling in digesting
the feedback, few of them really felt discouraged.

Finally, students were asked about how relevant it would be to integrate some
new functionalities in Café 2.0. From Fig. 12, it can be noticed that a large
amount of respondents (78%) would like to get more transparency regarding the
mistakes they make. It suggests that students may miss self-assessment skills
[31]. In the same vein, many students would appreciate to visualize their
actual progress with respect to what has been taught in classroom activities
so far. It may help them in self-regulating their time by being aware of where
their actually stand with respect to the course expectations. Next, 45% of
respondents expressed they strongly need the GameCodes to be integrated in
Café 2.0, which is fully consistent with the proportion of students whose
learning got boosted by the GameCodes (see Fig. 10). That need of digitalising
the GameCodes makes sense as, currently, the GameCodes are presented as
interactive PDF files where the different pages represent the resolution step,
which results in heavy documents and low quality-of-experience. Due to its
dynamics nature, the GameCodes should definitely be transposed in an only
plateform where the content can be organised so that the student easily access
the information they need, the rest being fully hidden.

Figure 10: Perception of the Pca with respect to other activities. Figure 11:
Perception of the feedback. Figure 12: Need for new functionalities.

####  IV-B2 Participation

Fig. 13a depicts an upset plot [32] illustrating how students participated to
the Pca. The figure is made up of three parts: the matrix (bottom right) shows
to participation. A dot in the matrix means that at least one student has
participated to that Challenge. If there are multiple black dots on a column,
it corresponds to students having participated to multiple Challenges (e.g.,
the first column refers to students having participated to the six
Challenges). The histogram on the bottom left is the size of each matrix row,
while the histogram on top right gives the number of students in the
corresponding column of the matrix. We see that Challenge 1 was the most
carried out (82% of registered students) but the number of students dropped
over time. This is normal in our country as we face a large attrition rate
during the first semester for new comers. Only 23 students took the six
Challenges (23.8% of registered students).

(a) Participation to the Challenges.

(b) Accessing Café 2.0 during Academic Year 2022–2023.

Figure 13: Results on participation.

Then, Fig 13b distills students’ participation over time by highlighting, for
each day of the semester, the number of students who started a session on Café
2.0. More specifically, the two red squares mark the two main evaluations of
the course : the midterm and the final exam. Next, the black rectangles refer
to the certificative periods of the Pca. Knowing that, we can note that the
number of sessions consistently concentrate during those periods and just
before the exam. We can also notice a higher number of sessions a few days
before the midterm. The days in-between were dedicated to midterms related to
other courses, that is likely why few or no session got launched those days.
More generally, it appears that students get the most active only when they
are against some evaluation rather than regularly practicing to master what
they are being taught. That statement corroborates many research results [33,
34, 23] stating that students have difficulties to self-regulate. However, we
can still notice few connections spread along the semester, reflecting
students who autonomously took benefits from Café 2.0.

##  V Extending Café 2.0

In Sec. II, Café 2.0 was introduced as an interdisciplinary learning tool,
aiming to train abstraction and problem-solving skills in general. This
section consolidates that ambition by detailing the preparation to perform in
order to fit Café 2.0 with a new problem profile, related to a STEM field.

###  V-A General Consideration

To integrate a new problem profile in Café 2.0, the following requirements
must be met :

  1. Requirement 1

: The resolution should be paved by a sequence of production steps.

  2. Requirement 2

: The resolution should run through two phases : the abstraction one and the
concrete one.

  3. Requirement 3

: The abstraction phase should rely on a graphical reasoning.

  4. Requirement 4

: The graphical representation should be dynamic, in such a way that it can be
manipulated to illustrate different solution states (general ones and specific
ones).

  5. Requirement 5

: The graphical representation should be made up of predefined graphical
components. They can stand as placeholders or movable elements students must
handle when they are designing a solution.

Besides this, independently from the discipline, an activity (that can be
similar to the Pca, described in Sec. III-A) requires to be set up to make
Café 2.0 standing as an integral part of the whole course activities. Finally,
it is worth noticing that having first year students as target public is the
most appropriate in order to avoid too complex solution modeling. Moreover, it
is likely that first year students are those who need this kind of support the
most.

###  V-B Application to Physics

To emphase the interdisciplinary potential of Café 2.0, we match it to a
specific problem profile picked from another field than Computer Science. In
particular, we are considering the following Kinematics problem in Physics :

A car of mass $m=1200kg$ is parked on a slope of $\alpha=30^{\circ}$. We would
like to compute the magnitude of the friction forces so that the car is at
rest.

(a) Resolution Flow in Physics

(b) Blank Diagram in Physics.

Figure 14: How Café 2.0 can fit to a Physics Introduction course.

####  V-B1 Activity Resolution Setting

Considering that specific problem type, five productions could be defined :
($i$) Representing the situation and identifying the forces that are applied
on the object of interest (being the car here) ; ($ii$) Choosing a system and
decomposing the different forces so that they follow the direction imposed by
the system ; ($iii$) Deriving the mathematical expression(s) allowing one to
formulate the friction forces ; ($iv$) Transposing the general representation
in a particular case (in which the problem might be possible to be intuitively
solved) ; ($v$) Using numerical data to compute the solution. Fig. 14a
illustrates those productions and map them to the two phases Café 2.0 is
supporting (Requirement 2). Also, Fig. 14a shows that the resolution is done
in sequence (Requirement 1). Further, the Force Diagram corresponds to
Requirement 3. Finally, that drawing is manipulated on step 4, corresponding
to Requirement 4.

####  V-B2 Correction and Feedback Setting

To enable automatic Correction and Feedback, a misconception library needs to
be defined and fed. That library should cover most of the mistakes students
may fall in across their resolution, as explained in Sec II-C and illustrated
in Fig. 6. For instance, considering our Kinematics problem, for the second
production where forces should be decomposed according to the system, one
typical mistake might be that the students did not direct all the arrows
according to the orthonormed system that is set. Another example of mistake
could be that the second equation does not reflect the drawing above.

####  V-B3 Production Modeling Setting

Similarly to the Blank Graphical Loop Invariant, it is relevant to hide from
the expected pictorial representation of the situation the key components. In
this way, students must identify and link them on their own while still having
benchmarks thanks to the provided canvas. Furthermore, on the supervisor’s
side, those components need to be modeled (by having a specific semantic and
relation with each other) in order to enable automatic personalized feedback.
For the Kinematics problem of interest, a relevant blank diagram could be the
one exposed through Fig. 14b. That figure relies on different kind of
graphical components, each of them being mapped to a specific color. Like in
the Blank Graphical Loop Invariant, red boxes should host variables while
green ones expect some description picked from a dropdown list. In addition to
them, purple boxes stand for forces and movable arrows are illustrated in
yellow. All the rest is fixed.

####  V-B4 Drawing Editor Setting

As suggested above, the graphical components that would be useful to formally
represent a situation in Kinematics (Requirement 5) are variables, forces,
predefined descriptions, and movable arrows. In addition to them, some
relevant patterns should be identified and designed. Then, they should be
integrated in the Drawing Editor so that, once a teacher or a student wants to
depict a scenario in Kinematics, they can simply select a suitable pattern and
drag and drop the graphical components to build up their pictorial
representation.

##  VI Related Work

Many automated system providing programming exercises were already proposed
(e.g., [11, 12, 13, 14, 15, 16, 17]). Most of them apply test-based
correction, i.e., student’s code is corrected through unit testing (except
UNLOCK [16] that tackles the problem solving skills in general, not just
coding skills). WebCAT [11] even makes students write their own tests too.
Kumar’s Problets [12] enables step by step code execution as part of the
feedback. Closer to Café 2.0, Dodona [18] proposes programming assignments
with automated feedback and harness the data to regulate the teaching
materials. However, Dodona is specialized in practicing coding (considering
different programming languages) in a collaborative environment (by allowing
students to ask questions on a forum) while Café 2.0 focuses on students’
abstract thinking, upstream to their code. Other tools also offer some
features depicting an abstract representation of some pieces of code. Among
them, Virtual-C IDE [19] typically allows to visualize a C-program behavior.
Online Python tutor [35] allows students to execute their code step-by-step
and visualize the runtime state of each structure. And JASM.IN [36]
illustrates dynamic program state and provides state transition rules for the
execution of Java language constructs. Café 2.0 differentiates from those
three tools by guiding students in modeling their solution by themselves, so
that they can rely on it to build their code. On the contrary, Virtual-C IDE
and JASM.IN automatically post-represent the solution based on some pieces of
code students have submitted. Considering now the tools where students are
designing their solution on their own, you can namely find Python turtle
graphic library that was already demonstrated as improving abstraction skills
[37]. However, Café 2.0 goes further than that as it bridges students’
graphical representation and resulting code by checking the consistency
between those two versions of the solution and provides automatic feedback
with respect to that.

Automatic feedback has been extensively motivated, described, and discussed in
previous studies [38, 39]. In particular, Keuning et al. [39] have shown that
it is not that easy to tune feedback with respect to some specific needs
(proper to the topic and the statement). If we position Café 2.0 with respect
to feedback theoretical aspects, Café 2.0 implements “Answer-until-correct”
(AUC) [40] feedback, as students can refresh their solution as much as they
need to. It is similar to Singh et al. [41] where students get a numerical
value (the number of required changes) and the suggestion(s) on how to correct
the mistake(s). In addition, Café 2.0 meets the principles Nicol introduced
around student’s engagement, self-regulation, and academic experience (the
only missing dimension being the social one) [42].

Finally, regarding the methodology Café 2.0 is currently supporting, other
studies promoted problem-solving through predefined steps [43]. Furthermore,
the relevance of transiting through an abstract representation of a solution
was corroborated to prevent students from getting overwhelmed with specific
problem instances [44].

##  VII Conclusion

This paper describes Café 2.0, a tool whose purpose is to make students
regularly and activily work in order to maintain them on a correct track, by
providing instantaneous personalized feedback. In practice, Café 2.0 supports
online activities that are made up of problems to solve. For each new
statement, besides its definition, its corresponding solution needs to be
outlined and configured by a supervisor. In particular, the solution should be
articulated by different types of productions, each of them being framed
through a canvas with specific placeholders whose semantic and potential
resulting typical mistakes should be parametrized beforehand. This way, on the
one hand, students get guided across their resolution and, on the other hand,
automatic and personalized review gets enabled since student’s solutions can
be anticipated.

Café 2.0 is relevant to be integrated in any course where abstract
representation through drawings stands as the ground for constructing a
solution. More specifically, in our CS1 course, abstraction is introduced in
the context of building a loop piece of code. More precisely, the Graphical
Loop Invariant Based Programming methodology is taught. Therefore, the central
kind of canvas in a resolution flow is the Blank Graphical Loop Invariant,
representing the shape of a Graphical Loop Invariant with different kinds of
boxes to fill and movable bars, allowing to visualize the solution under
construction at a specific iteration.

As further work, Café 2.0 could be transposed in other disciplines dedicated
to first year students. The first steps towards such an extension have been
already taken in the context of a Physics course.

Besides this, more specifically, Café 2.0’s set of functionalities keeps being
expanded in terms of activities and data processing. First, regarding the
activities, in addition to the Pca, the GameCodes (Sec. III) are under
implementation. Next, on the long run, it is aimed to implement a third kind
of activity : the Cdb activity [45]. The purpose of this activity drastically
differs from the Pca and the GameCode as it targets to motivate our
programming methodology by applying it in a real-life scenario rather than
academically training it. Despite that differentiation, it remains very
relevant to integrate such an activity in Café 2.0 as it also relies on a
sequence of productions to submit, in response to a given problem. The only
difference is that, for each production, students can get more freedom since
the resulting “instantaneous” feedback is built by students reviewers rather
than a machine [46]. Therefore, the difficulty in putting in place such an
activity is shifted from the automatic feedback configuration to the peer-
feedback reviewing process where teams should be defined, time should be
punctuated, feedback should be supervised, and productions should move between
students. Besides the activities, it is also planned to dedicate a focus on
the Progress Tracker and the Learning Analytics Dashboard in order to get more
transparency about students’ learning activity and wisely adapt the content
students should practice.

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|  Géraldine Brieven is a Ph.D. student in Computer Science Education at the
University of Liège since 2021. She got a master’s degree in Civil Engineering
in Computer Science at the same University, in 2019. Before starting her
Ph.D., she worked for two years in an IT Consultancy company. Her research
interests are about Collaborative Learning and Computer-Assisted Learning
(relying on Automatic Feedback). It aims to motivate and train problem-solving
skills.
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|  Lev Malcev finished his bachelor’s degree in Computer Science at the
University of Liège in 2021. He is currently working as a freelance developer
and finishing his master’s degree in Computer Security. He is also involved in
developing Café 2.0, a tool giving students the opportunity to train their
problem-solving skills and allowing to collect learning analytics to study
students’ learning behavior.
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|  Benoit Donnet received his Ph.D. degree in Computer Science from the
Université Pierre et Marie Curie in 2006 and has been a PostDoc until 2011 at
the Université catholique de Louvain (Belgium). Mr. Donnet joined the
Montefiore Institute at the Université de Liège since 2011 where he was
appointed successively as Assistant Professor and Associate Professor. His
research interests are about Internet measurements (measurements scalability,
Internet topology discovery, measurements applied to security), network
modeling, middleboxes, new Internet architectures (LISP, Segment Routing), and
Computer Science Education.
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